SherbyNet - /page/wikibook/entry.mu

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Thomae's function

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Thomae's function is a <a href="Real_number" title="Real number">real</a>-valued <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of a real variable that can be defined as:<a href="#cite_note-Beanland-1">[1]</a>: 531

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<mtext> is rational), with </mtext>

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<mtext> coprime</mtext>

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<mtext> is irrational.</mtext>

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<annotation encoding="application/x-tex">{\displaystyle f(x)={\begin{cases}{\frac {1}{q}}&{\text{if }}x={\tfrac {p}{q}}\quad (x{\text{ is rational), with }}p\in \mathbb {Z} {\text{ and }}q\in \mathbb {N} {\text{ coprime}}\\0&{\text{if }}x{\text{ is irrational.}}\end{cases}}}</annotation>

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It is named after <a href="Carl_Johannes_Thomae" title="Carl Johannes Thomae">Carl Johannes Thomae</a>, but has many other names: the popcorn function, the raindrop function, the countable cloud function, the modified <a href="Dirichlet_function" title="Dirichlet function">Dirichlet function</a>, the ruler function (not to be confused with the integer <a href="Ruler_function" title="Ruler function">ruler function</a>),<a href="#cite_note-2">[2]</a> the Riemann function, or the Stars over Babylon (<a href="John_Horton_Conway" title="John Horton Conway">John Horton Conway</a>'s name).<a href="#cite_note-3">[3]</a> Thomae mentioned it as an example for an integrable function with infinitely many discontinuities in an early textbook on Riemann's notion of integration.<a href="#cite_note-Thomae-4">[4]</a>

Since every <a href="Rational_number" title="Rational number">rational number</a> has a unique representation with <a href="Coprime_integers" title="Coprime integers">coprime</a> (also termed relatively prime) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {Z} }</annotation>

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</math><img src="./d29b9c8775da833d554991bfefdfc54a53085728.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.65ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {Z} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle q\in \mathbb {N} }</annotation>

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</math><img src="./b67f03c27aecd54c188bc7c10153a0d56bdd516f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.588ex; height:2.509ex;" alt="{\displaystyle q\in \mathbb {N} }" loading="lazy">, the function is <a href="Well-defined" class="mw-redirect" title="Well-defined">well-defined</a>. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=+1}">

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<annotation encoding="application/x-tex">{\displaystyle q=+1}</annotation>

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</math><img src="./ad1378bb7aedeaa17e92fcd64c2167c2830812da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.139ex; height:2.509ex;" alt="{\displaystyle q=+1}" loading="lazy"> is the only number in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

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</math><img src="./fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy"> that is coprime to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=0.}">

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<annotation encoding="application/x-tex">{\displaystyle p=0.}</annotation>

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</math><img src="./60fe1fffab8e03e1fad67889b59b90d095c29481.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.167ex; height:2.509ex;" alt="{\displaystyle p=0.}" loading="lazy">

It is a modification of the <a href="Dirichlet_function" title="Dirichlet function">Dirichlet function</a>, which is 1 at rational numbers and 0 elsewhere.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<div><ul><li>Thomae's function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is <a href="Bounded_function" title="Bounded function">bounded</a> and maps all real numbers to the <a href="Unit_interval" title="Unit interval">unit interval</a>:<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to [0,1].}">

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<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to [0,1].}</annotation>

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</math><img src="./3421513b7ab846170c65f380fbf5d5104f71be29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.807ex; height:2.843ex;" alt="{\displaystyle f:\mathbb {R} \to [0,1].}" loading="lazy"></li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is <a href="Periodic_function" title="Periodic function">periodic</a> with period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1:\;f(x+n)=f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle 1:\;f(x+n)=f(x)}</annotation>

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</math><img src="./b5a743a5c8bb599a9a2b041cd6ccade981558919.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.913ex; height:2.843ex;" alt="{\displaystyle 1:\;f(x+n)=f(x)}" loading="lazy"> for all <a href="Integer" title="Integer">integers</a> n and all real x.

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<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof of periodicity</div>

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For all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} \setminus \mathbb {Q} ,}">

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<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} \setminus \mathbb {Q} ,}</annotation>

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</math><img src="./05c36df9d57f17d242f3e83134ce2cf70d47d7c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.498ex; height:2.843ex;" alt="{\displaystyle x\in \mathbb {R} \setminus \mathbb {Q} ,}" loading="lazy"> we also have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+n\in \mathbb {R} \setminus \mathbb {Q} }">

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<annotation encoding="application/x-tex">{\displaystyle x+n\in \mathbb {R} \setminus \mathbb {Q} }</annotation>

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</math><img src="./d85d0ea0e1423dc03dfbfe645e08641936c34762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.086ex; height:2.843ex;" alt="{\displaystyle x+n\in \mathbb {R} \setminus \mathbb {Q} }" loading="lazy"> and hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x+n)=f(x)=0,}">

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<annotation encoding="application/x-tex">{\displaystyle f(x+n)=f(x)=0,}</annotation>

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</math><img src="./9b7e99b701fb3d1f35a67654ed3d27ef072ed0ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.076ex; height:2.843ex;" alt="{\displaystyle f(x+n)=f(x)=0,}" loading="lazy">

For all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {Q} ,\;}">

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<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {Q} ,\;}</annotation>

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</math><img src="./8c69f51c9fccd7074e9ffd250a8cd64ccb647a98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.27ex; height:2.509ex;" alt="{\displaystyle x\in \mathbb {Q} ,\;}" loading="lazy"> there exist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {Z} }</annotation>

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</math><img src="./d29b9c8775da833d554991bfefdfc54a53085728.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.65ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {Z} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle q\in \mathbb {N} }</annotation>

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</math><img src="./b67f03c27aecd54c188bc7c10153a0d56bdd516f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.588ex; height:2.509ex;" alt="{\displaystyle q\in \mathbb {N} }" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;x=p/q,\;}">

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<annotation encoding="application/x-tex">{\displaystyle \;x=p/q,\;}</annotation>

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</math><img src="./419ff16c560f6f6ee2db36f3d14eb1e3863e7002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.767ex; height:2.843ex;" alt="{\displaystyle \;x=p/q,\;}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(p,\;q)=1.}">

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<annotation encoding="application/x-tex">{\displaystyle \gcd(p,\;q)=1.}</annotation>

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</math><img src="./a4782576b257f11a10d1c481d6d1688779725356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.123ex; height:2.843ex;" alt="{\displaystyle \gcd(p,\;q)=1.}" loading="lazy">

Consider <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+n=(p+nq)/q}">

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<annotation encoding="application/x-tex">{\displaystyle x+n=(p+nq)/q}</annotation>

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</math><img src="./69e42995065a4797f8a33b689b8ee364ad8a1a38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.179ex; height:2.843ex;" alt="{\displaystyle x+n=(p+nq)/q}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

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</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> <a href="Divisor" title="Divisor">divides</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">, it divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+nq}">

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<annotation encoding="application/x-tex">{\displaystyle p+nq}</annotation>

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</math><img src="./ed1a387dd7822bdbc1c71459c821bce41d6d3837.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.564ex; height:2.343ex;" alt="{\displaystyle p+nq}" loading="lazy">

and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">. Conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

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</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+nq}">

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<annotation encoding="application/x-tex">{\displaystyle p+nq}</annotation>

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</math><img src="./ed1a387dd7822bdbc1c71459c821bce41d6d3837.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.564ex; height:2.343ex;" alt="{\displaystyle p+nq}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">, it divides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p+nq)-nq=p}">

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<annotation encoding="application/x-tex">{\displaystyle (p+nq)-nq=p}</annotation>

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</math><img src="./094f9920e9a177e7ebc114d123358cd15cc31cca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.856ex; height:2.843ex;" alt="{\displaystyle (p+nq)-nq=p}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">. So <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(p+nq,q)=\gcd(p,q)=1}">

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<annotation encoding="application/x-tex">{\displaystyle \gcd(p+nq,q)=\gcd(p,q)=1}</annotation>

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</math><img src="./eabc5e701b9fe871c3386d6c2247cac029d31b89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.803ex; height:2.843ex;" alt="{\displaystyle \gcd(p+nq,q)=\gcd(p,q)=1}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x+n)=1/q=f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x+n)=1/q=f(x)}</annotation>

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</math><img src="./cf523fecfa7f7782b8705a044f41aaa97a6ddeaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.662ex; height:2.843ex;" alt="{\displaystyle f(x+n)=1/q=f(x)}" loading="lazy">.

</td></tr></tbody></table></div></li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is <a href="Continuous_function" title="Continuous function">discontinuous</a> at every rational number, so its points of discontinuity are <a href="Dense_set" title="Dense set">dense</a> within the real numbers.

<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof of discontinuity at rational numbers</div>

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Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=p/q}">

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<annotation encoding="application/x-tex">{\displaystyle x_{0}=p/q}</annotation>

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</math><img src="./d579beec02e8f7e7c235bb7583dfaf63ddb79b75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.884ex; height:2.843ex;" alt="{\displaystyle x_{0}=p/q}" loading="lazy"> be an arbitrary rational number, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;p\in \mathbb {Z} ,\;q\in \mathbb {N} ,}">

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<annotation encoding="application/x-tex">{\displaystyle \;p\in \mathbb {Z} ,\;q\in \mathbb {N} ,}</annotation>

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</math><img src="./5f2995d9b324af0ea0abbe37c2dadcf9727095da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.12ex; height:2.509ex;" alt="{\displaystyle \;p\in \mathbb {Z} ,\;q\in \mathbb {N} ,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> coprime.

This establishes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{0})=1/q.}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{0})=1/q.}</annotation>

</semantics>

</math><img src="./acf2d3fc86bcb1b88f3dbb33a2d34a028dc6dcdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.612ex; height:2.843ex;" alt="{\displaystyle f(x_{0})=1/q.}" loading="lazy">

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\alpha \in \mathbb {R} \setminus \mathbb {Q} \;}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;\alpha \in \mathbb {R} \setminus \mathbb {Q} \;}</annotation>

</semantics>

</math><img src="./8902b5f0fa2884694211d00f1641b5413b88980e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.3ex; height:2.843ex;" alt="{\displaystyle \;\alpha \in \mathbb {R} \setminus \mathbb {Q} \;}" loading="lazy"> be any <a href="Irrational_number" title="Irrational number">irrational number</a> and define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}=x_{0}+{\frac {\alpha }{n}}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}=x_{0}+{\frac {\alpha }{n}}}</annotation>

</semantics>

</math><img src="./3caa006c54998ac39b2178472e5356c6472191c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.195ex; height:4.676ex;" alt="{\displaystyle x_{n}=x_{0}+{\frac {\alpha }{n}}}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} .}</annotation>

</semantics>

</math><img src="./f6b82f4024a32a795b10f00971e03ad88ebaa367.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.56ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} .}" loading="lazy">

These <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation>

</semantics>

</math><img src="./7c5ea190699149306d242b70439e663559e3ffbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" loading="lazy"> are all irrational, and so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{n})=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x_{n})=0}</annotation>

</semantics>

</math><img src="./540ec6ab02b1b8ceb795f09942a04a8f54cc2c75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.897ex; height:2.843ex;" alt="{\displaystyle f(x_{n})=0}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} .}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} .}</annotation>

</semantics>

</math><img src="./f6b82f4024a32a795b10f00971e03ad88ebaa367.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.56ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} .}" loading="lazy">

This implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x_{0}-x_{n}|={\frac {\alpha }{n}},}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |x_{0}-x_{n}|={\frac {\alpha }{n}},}</annotation>

</semantics>

</math><img src="./873ac14726c8f1682be18a121072460fea170810.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.135ex; height:4.676ex;" alt="{\displaystyle |x_{0}-x_{n}|={\frac {\alpha }{n}},}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(x_{0})-f(x_{n})|={\frac {1}{q}}.}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |f(x_{0})-f(x_{n})|={\frac {1}{q}}.}</annotation>

</semantics>

</math><img src="./af69cf5beaf39d199da1787a6dc6ace50d99e762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.986ex; height:5.676ex;" alt="{\displaystyle |f(x_{0})-f(x_{n})|={\frac {1}{q}}.}" loading="lazy">

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\varepsilon =1/q\;}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;\varepsilon =1/q\;}</annotation>

</semantics>

</math><img src="./6fee333b5c281353cf2dde657f86dd372e882a7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.867ex; height:2.843ex;" alt="{\displaystyle \;\varepsilon =1/q\;}" loading="lazy">, and given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation>

</semantics>

</math><img src="./595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"> let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1+\left\lceil {\frac {\alpha }{\delta }}\right\rceil .}">

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=1+\left\lceil {\frac {\alpha }{\delta }}\right\rceil .}</annotation>

</semantics>

</math><img src="./83b3685320b5d477764a069507076a273b6b2c51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.309ex; height:5.009ex;" alt="{\displaystyle n=1+\left\lceil {\frac {\alpha }{\delta }}\right\rceil .}" loading="lazy"> For the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;x_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;x_{n}}</annotation>

</semantics>

</math><img src="./09781254e12059635b0b1fde2fb247bbcf9e1ba6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.193ex; height:2.009ex;" alt="{\displaystyle \;x_{n}}" loading="lazy"> we have

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |f(x_{0})-f(x_{n})|=1/q\geq \varepsilon }</annotation>

</semantics>

</math> and

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |x_{0}-x_{n}|={\frac {\alpha }{n}}={\frac {\alpha }{1+\left\lceil {\frac {\alpha }{\delta }}\right\rceil }}<{\frac {\alpha }{\left\lceil {\frac {\alpha }{\delta }}\right\rceil }}\leq \delta ,}</annotation>

</semantics>

</math>

which is exactly the definition of discontinuity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy">.

</td></tr></tbody></table></div></li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous at every <a href="Irrational_number" title="Irrational number">irrational number</a>, so its points of continuity are <a href="Dense_set" title="Dense set">dense</a> within the real numbers.

<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof of continuity at irrational arguments</div>

</th></tr>

<tr>

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is periodic with period <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in \mathbb {Q} ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\in \mathbb {Q} ,}</annotation>

</semantics>

</math><img src="./2b042eb625e3ce198cea3fdbc8a4f502f3ac9162.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.458ex; height:2.509ex;" alt="{\displaystyle 0\in \mathbb {Q} ,}" loading="lazy"> it suffices to check all irrational points in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=(0,1).\;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I=(0,1).\;}</annotation>

</semantics>

</math><img src="./7b4c462d94a1b65bf735c783fe770e70415b5298.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.73ex; height:2.843ex;" alt="{\displaystyle I=(0,1).\;}" loading="lazy"> Assume now <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0,\;i\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0,\;i\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./c373c28bd536824015f2c7bed9def04a68c2135d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.345ex; height:2.509ex;" alt="{\displaystyle \varepsilon >0,\;i\in \mathbb {N} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in I\setminus \mathbb {Q} .}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}\in I\setminus \mathbb {Q} .}</annotation>

</semantics>

</math><img src="./972162fb2231dfdd635ef363d7ba2fc9df69732b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.046ex; height:2.843ex;" alt="{\displaystyle x_{0}\in I\setminus \mathbb {Q} .}" loading="lazy"> According to the <a href="Archimedean_property" title="Archimedean property">Archimedean property</a> of the reals, there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./2f7b1b20ccd888e84ebffce5446840cc882b3dc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.567ex; height:2.176ex;" alt="{\displaystyle r\in \mathbb {N} }" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/r<\varepsilon ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/r<\varepsilon ,}</annotation>

</semantics>

</math><img src="./c6592511b52506448f60054202234cce7159a504.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.202ex; height:2.843ex;" alt="{\displaystyle 1/r<\varepsilon ,}" loading="lazy"> and there exist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;k_{i}\in \mathbb {N} ,}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;k_{i}\in \mathbb {N} ,}</annotation>

</semantics>

</math><img src="./3987709a6cb11673135196b6fe243d0e833fb071.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.822ex; height:2.509ex;" alt="{\displaystyle \;k_{i}\in \mathbb {N} ,}" loading="lazy"> such that

for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\ldots ,r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=1,\ldots ,r}</annotation>

</semantics>

</math><img src="./2be33d7fc67e3a9c77e1fd1e94f0c21160221988.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.29ex; height:2.509ex;" alt="{\displaystyle i=1,\ldots ,r}" loading="lazy"> we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<{\frac {k_{i}}{i}}<x_{0}<{\frac {k_{i}+1}{i}}.}">

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0<{\frac {k_{i}}{i}}<x_{0}<{\frac {k_{i}+1}{i}}.}</annotation>

</semantics>

</math><img src="./51fc2a9f96826cd5d0be1b4df8c1e85bd26c3002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.186ex; height:5.343ex;" alt="{\displaystyle 0<{\frac {k_{i}}{i}}<x_{0}<{\frac {k_{i}+1}{i}}.}" loading="lazy">

The minimal distance of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> to its i-th lower and upper bounds equals

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{i}:=\min \left\{\left|x_{0}-{\frac {k_{i}}{i}}\right|,\;\left|x_{0}-{\frac {k_{i}+1}{i}}\right|\right\}.}</annotation>

</semantics>

</math>

We define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

</semantics>

</math><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> as the minimum of all the finitely many <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{i}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{i}.}</annotation>

</semantics>

</math><img src="./a0a0fcf80f045e67f9c726c5bba64d9e009c3fa7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.655ex; height:2.509ex;" alt="{\displaystyle d_{i}.}" loading="lazy">

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta :=\min _{1\leq i\leq r}\{d_{i}\},\;}</annotation>

</semantics>

</math> so that

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1,\dots ,r,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=1,\dots ,r,}</annotation>

</semantics>

</math><img src="./ff6c4dc3fe436b21da62f7b80d0321f712b7b12a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.937ex; height:2.509ex;" alt="{\displaystyle i=1,\dots ,r,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x_{0}-k_{i}/i|\geq \delta }">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |x_{0}-k_{i}/i|\geq \delta }</annotation>

</semantics>

</math><img src="./496db0982e91ca748928907625df8d69631acdfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.641ex; height:2.843ex;" alt="{\displaystyle |x_{0}-k_{i}/i|\geq \delta }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x_{0}-(k_{i}+1)/i|\geq \delta .}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |x_{0}-(k_{i}+1)/i|\geq \delta .}</annotation>

</semantics>

</math><img src="./29ae06817ea256f64501bc08b297c3e128818471.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.1ex; height:2.843ex;" alt="{\displaystyle |x_{0}-(k_{i}+1)/i|\geq \delta .}" loading="lazy">

This is to say, all these rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}/i,\;(k_{i}+1)/i,\;}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k_{i}/i,\;(k_{i}+1)/i,\;}</annotation>

</semantics>

</math><img src="./05b9f25e9903a8fa7658ec98a32afa584e713746.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.735ex; height:2.843ex;" alt="{\displaystyle k_{i}/i,\;(k_{i}+1)/i,\;}" loading="lazy"> are outside the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

</semantics>

</math><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy">-neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}.}</annotation>

</semantics>

</math><img src="./7ec756993d89cc1bd74f84040c07f5e11f0a8102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.031ex; height:2.009ex;" alt="{\displaystyle x_{0}.}" loading="lazy">

Now let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {Q} \cap (x_{0}-\delta ,x_{0}+\delta )}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {Q} \cap (x_{0}-\delta ,x_{0}+\delta )}</annotation>

</semantics>

</math><img src="./c9bf704eb184bbdaaf7127d294efae33f7264bef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.95ex; height:2.843ex;" alt="{\displaystyle x\in \mathbb {Q} \cap (x_{0}-\delta ,x_{0}+\delta )}" loading="lazy"> with the unique representation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=p/q}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=p/q}</annotation>

</semantics>

</math><img src="./6424573baa7450dcdea46454ec43e9e6817b27c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.83ex; height:2.843ex;" alt="{\displaystyle x=p/q}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p,q\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./e442b40aed58eb2de70076c1580f56d24022b83c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.881ex; height:2.509ex;" alt="{\displaystyle p,q\in \mathbb {N} }" loading="lazy"> are coprime. Then, necessarily, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q>r,\;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q>r,\;}</annotation>

</semantics>

</math><img src="./b032237be0c502e9b0121e8902b0612926682b62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.509ex; height:2.176ex;" alt="{\displaystyle q>r,\;}" loading="lazy"> and therefore,

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=1/q<1/r<\varepsilon .}</annotation>

</semantics>

</math>

Likewise, for all irrational <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in I,\;f(x)=0=f(x_{0}),\;}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in I,\;f(x)=0=f(x_{0}),\;}</annotation>

</semantics>

</math><img src="./f8ae65a38ceed759cee0649d3b10287d8d154779.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.562ex; height:2.843ex;" alt="{\displaystyle x\in I,\;f(x)=0=f(x_{0}),\;}" loading="lazy"> and thus, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation>

</semantics>

</math><img src="./e04ec3670b50384a3ce48aca42e7cc5131a06b12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}" loading="lazy"> then any choice of (sufficiently small) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta >0}</annotation>

</semantics>

</math><img src="./595d5cea06fdcaf2642caf549eda2cfc537958a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.343ex;" alt="{\displaystyle \delta >0}" loading="lazy"> gives

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |x-x_{0}|<\delta \implies |f(x_{0})-f(x)|=f(x)<\varepsilon .}</annotation>

</semantics>

</math>

Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is continuous on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \setminus \mathbb {Q} .}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \setminus \mathbb {Q} .}</annotation>

</semantics>

</math><img src="./68fc162257d6dd39abcde0d8d66908c865086cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.328ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \setminus \mathbb {Q} .}" loading="lazy">

</td></tr></tbody></table></div></li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is nowhere differentiable.

<th class="cot-header-mainspace" style="; font-size:87%; padding:0.2em 0.3em; text-align:center;"><div style="font-size:115%;margin:0 4em">Proof of being nowhere differentiable</div>

</th></tr>

<tr>

<ul><li>For rational numbers, this follows from non-continuity.</li>

<li>For irrational numbers:

<dl><dd>For any <a href="Sequence" title="Sequence">sequence</a> of irrational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})_{n=1}^{\infty }}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a_{n})_{n=1}^{\infty }}</annotation>

</semantics>

</math><img src="./548835a1d47d9829b180e1deda337ebf1bade908.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.577ex; height:3.009ex;" alt="{\displaystyle (a_{n})_{n=1}^{\infty }}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}\neq x_{0}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{n}\neq x_{0}}</annotation>

</semantics>

</math><img src="./85711f4f4f23e72a3bd4c08fb7e2ec18d1f98db2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.931ex; height:2.676ex;" alt="{\displaystyle a_{n}\neq x_{0}}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} _{+}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} _{+}}</annotation>

</semantics>

</math><img src="./992c1534d4a61acc9ec8c7e79ed97b64f7fd01f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.424ex; height:2.509ex;" alt="{\displaystyle n\in \mathbb {N} _{+}}" loading="lazy"> that converges to the irrational point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy">, the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f(a_{n}))_{n=1}^{\infty }}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f(a_{n}))_{n=1}^{\infty }}</annotation>

</semantics>

</math><img src="./6d3984fdd2833d23ec17a98cbb46b884853044f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.664ex; height:3.009ex;" alt="{\displaystyle (f(a_{n}))_{n=1}^{\infty }}" loading="lazy"> is identically <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">, and so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }\left|{\frac {f(a_{n})-f(x_{0})}{a_{n}-x_{0}}}\right|=0}">

</mrow>

</munder>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }\left|{\frac {f(a_{n})-f(x_{0})}{a_{n}-x_{0}}}\right|=0}</annotation>

</semantics>

</math><img src="./5a2f5e1d14e8eac8f5b565bd72416d8d537ea6fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.899ex; height:6.509ex;" alt="{\displaystyle \lim _{n\to \infty }\left|{\frac {f(a_{n})-f(x_{0})}{a_{n}-x_{0}}}\right|=0}" loading="lazy">.</dd>

<dd>On the other hand, consider the sequence of rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b_{n})_{n=1}^{\infty }}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (b_{n})_{n=1}^{\infty }}</annotation>

</semantics>

</math><img src="./898c0d733b741d9142771a226a0769213ef9abff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.344ex; height:3.009ex;" alt="{\displaystyle (b_{n})_{n=1}^{\infty }}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}=\lfloor nx_{0}\rfloor /n}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{n}=\lfloor nx_{0}\rfloor /n}</annotation>

</semantics>

</math><img src="./37bd29c2912c114143b3f01512f4647c63464b95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.715ex; height:2.843ex;" alt="{\displaystyle b_{n}=\lfloor nx_{0}\rfloor /n}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor nx_{0}\rfloor }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lfloor nx_{0}\rfloor }</annotation>

</semantics>

</math><img src="./5d4c3e14774d69f26990babfc6fe68b13314d285.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.843ex; height:2.843ex;" alt="{\displaystyle \lfloor nx_{0}\rfloor }" loading="lazy"> denotes the <a href="Floor_and_ceiling_functions" title="Floor and ceiling functions">floor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nx_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle nx_{0}}</annotation>

</semantics>

</math><img src="./f288bb013faa4bf86723613040dbd0b924ae3fdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.779ex; height:2.009ex;" alt="{\displaystyle nx_{0}}" loading="lazy">. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nx_{0}-1<\lfloor nx_{0}\rfloor \leq nx_{0}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle nx_{0}-1<\lfloor nx_{0}\rfloor \leq nx_{0}}</annotation>

</semantics>

</math><img src="./da3d5783448eb0e8b733dd12d358a1b6ad8110b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.6ex; height:2.843ex;" alt="{\displaystyle nx_{0}-1<\lfloor nx_{0}\rfloor \leq nx_{0}}" loading="lazy">, the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b_{n})_{n=1}^{\infty }}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (b_{n})_{n=1}^{\infty }}</annotation>

</semantics>

</math><img src="./898c0d733b741d9142771a226a0769213ef9abff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.344ex; height:3.009ex;" alt="{\displaystyle (b_{n})_{n=1}^{\infty }}" loading="lazy"> converges to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> using the <a href="Squeeze_theorem" title="Squeeze theorem">Squeeze theorem</a>. Also, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |b_{n}-x_{0}|=|\lfloor nx_{0}\rfloor /n-x_{0}|=|\lfloor nx_{0}\rfloor -nx_{0}|/n\leq 1/n}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |b_{n}-x_{0}|=|\lfloor nx_{0}\rfloor /n-x_{0}|=|\lfloor nx_{0}\rfloor -nx_{0}|/n\leq 1/n}</annotation>

</semantics>

</math><img src="./bbb34ad12be2fc8e9e686c75616d1b0065a94434.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.981ex; height:2.843ex;" alt="{\displaystyle |b_{n}-x_{0}|=|\lfloor nx_{0}\rfloor /n-x_{0}|=|\lfloor nx_{0}\rfloor -nx_{0}|/n\leq 1/n}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">.</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|{\frac {f(b_{n})-f(x_{0})}{b_{n}-x_{0}}}\right|\geq {\frac {1/n-0}{1/n}}=1}">

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|{\frac {f(b_{n})-f(x_{0})}{b_{n}-x_{0}}}\right|\geq {\frac {1/n-0}{1/n}}=1}</annotation>

</semantics>

</math><img src="./edc84c14ffe4c1953f203decf51b8c5d1472f592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.664ex; height:6.509ex;" alt="{\displaystyle \left|{\frac {f(b_{n})-f(x_{0})}{b_{n}-x_{0}}}\right|\geq {\frac {1/n-0}{1/n}}=1}" loading="lazy">. Therefore we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \liminf _{n\to \infty }\left|{\frac {f(b_{n})-f(x_{0})}{b_{n}-x_{0}}}\right|\geq 1\neq 0}">

</mrow>

</munder>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \liminf _{n\to \infty }\left|{\frac {f(b_{n})-f(x_{0})}{b_{n}-x_{0}}}\right|\geq 1\neq 0}</annotation>

</semantics>

</math><img src="./cd77147622668bd8cafa7245a10ee91eff488b2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.924ex; height:6.509ex;" alt="{\displaystyle \liminf _{n\to \infty }\left|{\frac {f(b_{n})-f(x_{0})}{b_{n}-x_{0}}}\right|\geq 1\neq 0}" loading="lazy"> and so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is not differentiable at any irrational number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy">.</dd></dl></li></ul>

</td></tr></tbody></table></div></li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> has a proper <a href="Maxima_and_minima" class="mw-redirect" title="Maxima and minima">local maximum</a> at each rational number, providing an example of a function with a dense set of proper local maxima.<a href="#cite_note-5">[5]</a>

See the proofs for continuity and discontinuity above for the construction of appropriate <a href="Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)">neighbourhoods</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> has maxima.</li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is <a href="Riemann_integrable" class="mw-redirect" title="Riemann integrable">Riemann integrable</a> on any interval and the integral evaluates to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> over any set.

The <a href="Lebesgue_integrability_condition" class="mw-redirect" title="Lebesgue integrability condition">Lebesgue criterion for integrability</a> states that a bounded function is Riemann integrable if and only if the set of all discontinuities has <a href="Lebesgue_measure" title="Lebesgue measure">measure zero</a>.<a href="#cite_note-6">[6]</a> Every <a href="Countability" class="mw-redirect" title="Countability">countable</a> subset of the real numbers - such as the rational numbers - has measure zero, so the above discussion shows that Thomae's function is Riemann integrable on any interval. The function's integral is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> over any set because the function is equal to zero <a href="Almost_everywhere" title="Almost everywhere">almost everywhere</a>.</li><li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\{\,(x,f(x)):x\in (0,1)\,\}\subset \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=\{\,(x,f(x)):x\in (0,1)\,\}\subset \mathbb {R} ^{2}}</annotation>

</semantics>

</math><img src="./2d08e1b7b8cecff1faf6ff4af0d9dace10d7ce7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.721ex; height:3.176ex;" alt="{\displaystyle G=\{\,(x,f(x)):x\in (0,1)\,\}\subset \mathbb {R} ^{2}}" loading="lazy"> is the graph of the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,1)}</annotation>

</semantics>

</math><img src="./c79c6838e423c1ed3c7ea532a56dc9f9dae8290b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,1)}" loading="lazy">, then the <a href="Minkowski%E2%80%93Bouligand_dimension" title="Minkowski–Bouligand dimension">box-counting dimension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4/3}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4/3}</annotation>

</semantics>

</math><img src="./f2f29a3dad63994c875d297399d10029b43eda57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 4/3}" loading="lazy">.<a href="#cite_note-7">[7]</a></li></ul></div>

<div class="mw-heading mw-heading2"><h2 id="Related_probability_distributions">Related probability distributions</h2></div>

Empirical probability distributions related to Thomae's function appear in <a href="DNA_sequencing" title="DNA sequencing">DNA sequencing</a>.<a href="#cite_note-Trifonov-8">[8]</a> The human genome is <a href="Diploid" class="mw-redirect" title="Diploid">diploid</a>, having two strands per chromosome. When sequenced, small pieces ("reads") are generated: for each spot on the genome, an integer number of reads overlap with it. Their ratio is a rational number, and typically distributed similarly to Thomae's function.

If pairs of positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m,n}</annotation>

</semantics>

</math><img src="./6568e95b6bf8f39b7fd2c9b52b7b00ee124c6250.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.469ex; height:2.009ex;" alt="{\displaystyle m,n}" loading="lazy"> are sampled from a distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n,m)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(n,m)}</annotation>

</semantics>

</math><img src="./d853550fb72817544280e7925c56006062002c75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.557ex; height:2.843ex;" alt="{\displaystyle f(n,m)}" loading="lazy"> and used to generate ratios <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=n/(n+m)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q=n/(n+m)}</annotation>

</semantics>

</math><img src="./22d3aca5ed9ef2d46f0ad8f444ba0703a6dd280d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.81ex; height:2.843ex;" alt="{\displaystyle q=n/(n+m)}" loading="lazy">, this gives rise to a distribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(q)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(q)}</annotation>

</semantics>

</math><img src="./800073b2bb60999661bcc6be4316220923612cc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.995ex; height:2.843ex;" alt="{\displaystyle g(q)}" loading="lazy"> on the rational numbers. If the integers are independent the distribution can be viewed as a <a href="Convolution" title="Convolution">convolution</a> over the rational numbers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle g(a/(a+b))=\sum _{t=1}^{\infty }f(ta)f(tb)}">

</mrow>

</mrow>

</mrow>

</munderover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle g(a/(a+b))=\sum _{t=1}^{\infty }f(ta)f(tb)}</annotation>

</semantics>

</math><img src="./dd9f14d6fc55ceccb4ae10667f6e49f4710b328d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.143ex; height:3.176ex;" alt="{\textstyle g(a/(a+b))=\sum _{t=1}^{\infty }f(ta)f(tb)}" loading="lazy">. Closed form solutions exist for <a href="Power-law" class="mw-redirect" title="Power-law">power-law</a> distributions with a cut-off. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(k)=k^{-\alpha }e^{-\beta k}/\mathrm {Li} _{\alpha }(e^{-\beta })}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(k)=k^{-\alpha }e^{-\beta k}/\mathrm {Li} _{\alpha }(e^{-\beta })}</annotation>

</semantics>

</math><img src="./22d5edef093ad32a049df6db8a91e69c655fbddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.456ex; height:3.176ex;" alt="{\displaystyle f(k)=k^{-\alpha }e^{-\beta k}/\mathrm {Li} _{\alpha }(e^{-\beta })}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Li} _{\alpha }}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Li} _{\alpha }}</annotation>

</semantics>

</math><img src="./e5e3e2271f42576b3c1deae78eb40f70e4918539.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.384ex; height:2.509ex;" alt="{\displaystyle \mathrm {Li} _{\alpha }}" loading="lazy"> is the <a href="Polylogarithm" title="Polylogarithm">polylogarithm</a> function) then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(a/(a+b))=(ab)^{-\alpha }\mathrm {Li} _{2\alpha }(e^{-(a+b)\beta })/\mathrm {Li} _{\alpha }^{2}(e^{-\beta })}">

</mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(a/(a+b))=(ab)^{-\alpha }\mathrm {Li} _{2\alpha }(e^{-(a+b)\beta })/\mathrm {Li} _{\alpha }^{2}(e^{-\beta })}</annotation>

</semantics>

</math><img src="./61fa89b125253f2005d0cda653d7d7f07528637a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.468ex; height:3.343ex;" alt="{\displaystyle g(a/(a+b))=(ab)^{-\alpha }\mathrm {Li} _{2\alpha }(e^{-(a+b)\beta })/\mathrm {Li} _{\alpha }^{2}(e^{-\beta })}" loading="lazy">. In the case of uniform distributions on the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots ,L\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots ,L\}}</annotation>

</semantics>

</math><img src="./d0323026725f54cf51da74aa680edcf75caed531.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.445ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,L\}}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(a/(a+b))=(1/L^{2})\lfloor L/\max(a,b)\rfloor }">

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(a/(a+b))=(1/L^{2})\lfloor L/\max(a,b)\rfloor }</annotation>

</semantics>

</math><img src="./0863915226364c74f068fdb1cf7483ce65c3b5eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.658ex; height:3.176ex;" alt="{\displaystyle g(a/(a+b))=(1/L^{2})\lfloor L/\max(a,b)\rfloor }" loading="lazy">, which is very similar to Thomae's function.<a href="#cite_note-Trifonov-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="The_ruler_function">The ruler function</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Ruler_function" title="Ruler function">Ruler function</a></div>

For integers, the exponent of the highest power of 2 dividing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> gives 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, ... (sequence <a href="https://oeis.org/A007814" class="extiw external" title="oeis:A007814">A007814</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). If 1 is added, or if the 0s are removed, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, ... (sequence <a href="https://oeis.org/A001511" class="extiw external" title="oeis:A001511">A001511</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). The values resemble tick-marks on a 1/16th <a href="Graduation_(scale)" title="Graduation (scale)">graduated ruler</a>, hence the name. These values correspond to the restriction of the Thomae function to the <a href="Dyadic_rational" title="Dyadic rational">dyadic rationals</a>: those rational numbers whose denominators are powers of 2.

<div class="mw-heading mw-heading2"><h2 id="Related_functions">Related functions</h2></div>

A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. The set of discontinuities of any function must be an <a href="F%CF%83_set" title="Fσ set">Fσ set</a>. If such a function existed, then the irrationals would be an Fσ set. The irrationals would then be the <a href="Countable_set" title="Countable set">countable</a> <a href="Union_(set_theory)" title="Union (set theory)">union</a> of <a href="Closed_set" title="Closed set">closed sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \bigcup _{i=0}^{\infty }C_{i}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \bigcup _{i=0}^{\infty }C_{i}}</annotation>

</semantics>

</math><img src="./1d3e207d898c85c68cf7e56aaec246f9e91d6bfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.685ex; height:3.176ex;" alt="{\textstyle \bigcup _{i=0}^{\infty }C_{i}}" loading="lazy">, but since the irrationals do not contain an interval, neither can any of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{i}}</annotation>

</semantics>

</math><img src="./cc49dc02c0ec8c86b67e7d10518ac791eda0bf22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.509ex;" alt="{\displaystyle C_{i}}" loading="lazy">. Therefore, each of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{i}}</annotation>

</semantics>

</math><img src="./cc49dc02c0ec8c86b67e7d10518ac791eda0bf22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.509ex;" alt="{\displaystyle C_{i}}" loading="lazy"> would be nowhere dense, and the irrationals would be a <a href="Meager_set" class="mw-redirect" title="Meager set">meager set</a>. It would follow that the real numbers, being the union of the irrationals and the rationals (which, as a countable set, is evidently meager), would also be a meager set. This would contradict the <a href="Baire_category_theorem" title="Baire category theorem">Baire category theorem</a>: because the reals form a <a href="Complete_metric_space" title="Complete metric space">complete metric space</a>, they form a <a href="Baire_space" title="Baire space">Baire space</a>, which cannot be meager in itself.

A variant of Thomae's function can be used to show that any Fσ subset of the real numbers can be the set of discontinuities of a function. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A=\bigcup _{n=1}^{\infty }F_{n}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A=\bigcup _{n=1}^{\infty }F_{n}}</annotation>

</semantics>

</math><img src="./ad640ce8aef3da5b3ba15fda2c80aea677694ccc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.197ex; height:3.176ex;" alt="{\textstyle A=\bigcup _{n=1}^{\infty }F_{n}}" loading="lazy"> is a countable union of closed sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy">, define

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mtd>

<mtd>

</mrow>

<mtext> is rational and </mtext>

</mrow>

<mtext> is minimal so that </mtext>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mtd>

<mtd>

</mrow>

<mtext> is irrational and </mtext>

</mrow>

<mtext> is minimal so that </mtext>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{A}(x)={\begin{cases}{\frac {1}{n}}&{\text{if }}x{\text{ is rational and }}n{\text{ is minimal so that }}x\in F_{n}\\-{\frac {1}{n}}&{\text{if }}x{\text{ is irrational and }}n{\text{ is minimal so that }}x\in F_{n}\\0&{\text{if }}x\notin A\end{cases}}}</annotation>

</semantics>

</math>

Then a similar argument as for Thomae's function shows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{A}}</annotation>

</semantics>

</math><img src="./f4df88573ffd17201a3c77be3bee49037c3f9aa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.604ex; height:2.509ex;" alt="{\displaystyle f_{A}}" loading="lazy"> has A as its set of discontinuities.

<ul><li><a href="Blumberg_theorem" title="Blumberg theorem">Blumberg theorem</a></li>

<li><a href="Cantor_function" title="Cantor function">Cantor function</a></li>

<li><a href="Dirichlet_function" title="Dirichlet function">Dirichlet function</a></li>

<li><a href="Euclid's_orchard" title="Euclid's orchard">Euclid's orchard</a> – Thomae's function can be interpreted as a perspective drawing of Euclid's orchard</li>

<li><a href="Volterra's_function" title="Volterra's function">Volterra's function</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBeanlandRobertsStevenson2009" class="citation journal cs1">Beanland, Kevin; Roberts, James W.; Stevenson, Craig (2009). "Modifications of Thomae's Function and Differentiability". <a href="The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>. 116 (6): 531–535. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2F193009709x470425">10.4169/193009709x470425</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/40391145">40391145</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFDunham2008" class="citation book cs1"><a href="William_Dunham_(mathematician)" title="William Dunham (mathematician)">Dunham, William</a> (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aYTYBQAAQBAJ&pg=PA149">The Calculus Gallery: Masterpieces from Newton to Lebesgue</a>. Princeton: Princeton University Press. page 149, chapter 10. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-13626-4</bdi>. <q>...the so-called ruler function, a simple but provocative example that appeared in a work of Johannes Karl Thomae ... The graph suggests the vertical markings on a ruler—hence the name.</q></cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFJohn_Conway" class="citation web cs1">John Conway. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180613235037/mathforum.org/kb/message.jspa?messageID=1375516">"Topic: Provenance of a function"</a>. The Math Forum. Archived from <a rel="nofollow" class="external text" href="http://mathforum.org/kb/message.jspa?messageID=1375516">the original</a> on 13 June 2018.</cite>

</li>

<li id="cite_note-Thomae-4"><a href="#cite_ref-Thomae_4-0">^</a> <cite id="CITEREFThomae1875" class="citation book cs1 cs1-prop-foreign-lang-source">Thomae, J. (1875). Einleitung in die Theorie der bestimmten Integrale (in German). Halle a/S: Verlag von Louis Nebert. p. 14, §20.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFPerfetti2006" class="citation journal cs1">Perfetti, Paolo (Fall 2006). "Solution to Problem 1129". Problem Department. Pi Mu Epsilon Journal. 12 (5): 301–319. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/24337958">24337958</a>.</cite> Perfetti supplies the negation of Thomae's function as an example with a dense set of proper local minima.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFSpivak1965" class="citation book cs1"><a href="Michael_Spivak" title="Michael Spivak">Spivak, M.</a> (1965). Calculus on manifolds. Perseus Books. page 53, Theorem 3-8. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8053-9021-6</bdi>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFChenFraserYu2022" class="citation journal cs1">Chen, Haipeng; Fraser, Jonathan M.; Yu, Han (2022). "Dimensions of the popcorn graph". <a href="Proceedings_of_the_American_Mathematical_Society" title="Proceedings of the American Mathematical Society">Proceedings of the American Mathematical Society</a>. 150 (11): 4729–4742. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2007.08407">2007.08407</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fproc%2F15729">10.1090/proc/15729</a>.</cite>

</li>

<li id="cite_note-Trifonov-8">^ <a href="#cite_ref-Trifonov_8-0">a</a> <a href="#cite_ref-Trifonov_8-1">b</a> <cite id="CITEREFTrifonovPasqualucciDalla-FaveraRabadan2011" class="citation journal cs1">Trifonov, Vladimir; Pasqualucci, Laura; Dalla-Favera, Riccardo; Rabadan, Raul (2011). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3240948">"Fractal-like Distributions over the Rational Numbers in High-throughput Biological and Clinical Data"</a>. Scientific Reports. 1 (191): 191. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1010.4328">1010.4328</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011NatSR...1E.191T">2011NatSR...1E.191T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fsrep00191">10.1038/srep00191</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3240948">3240948</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22355706">22355706</a>.</cite>

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<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<ul><li><cite id="CITEREFAbbott2016" class="citation cs1">Abbott, Stephen (2016). Understanding Analysis (Softcover reprint of the original 2nd ed.). New York: <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4939-5026-3</bdi>.</cite></li>

<li><cite id="CITEREFBartleSherbert1999" class="citation cs1">Bartle, Robert G.; Sherbert, Donald R. (1999). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontore00bart_1">Introduction to Real Analysis</a> (3rd ed.). Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-32148-4</bdi>.</cite> (Example 5.1.6 (h))</li></ul>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Dirichlet-function">"Dirichlet-function"</a>. <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>. <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>. 2001 [1994].</cite></li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DirichletFunction.html">"Dirichlet Function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-27" href="https://en.wikipedia.org/wiki/?title=Thomae's_function&oldid=1302739877">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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