SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Monotonic function</title>

</head>

Monotonic function

</h1>

</header>

/* 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">"Monotonicity" redirects here. For information on monotonicity as it pertains to <a href="Voting_systems" class="mw-redirect" title="Voting systems">voting systems</a>, see <a href="Monotonicity_criterion" class="mw-redirect" title="Monotonicity criterion">monotonicity criterion</a>. For information on monotonicity as it pertains to logical systems, see <a href="Monotonicity_of_entailment" title="Monotonicity of entailment">Monotonicity of entailment</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Monotonic" redirects here. For other uses, see <a href="Monotone_(disambiguation)" class="mw-redirect mw-disambig" title="Monotone (disambiguation)">Monotone (disambiguation)</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, a monotonic function (or monotone function) is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> between <a href="List_of_order_structures_in_mathematics" title="List of order structures in mathematics">ordered sets</a> that preserves or reverses the given <a href="Order_relation" class="mw-redirect" title="Order relation">order</a>.<a href="#cite_note-1">[1]</a><a href="#cite_note-:1-2">[2]</a><a href="#cite_note-:0-3">[3]</a> This concept first arose in <a href="Calculus" title="Calculus">calculus</a>, and was later generalized to the more abstract setting of <a href="Order_theory" title="Order theory">order theory</a>.

<div class="mw-heading mw-heading2"><h2 id="In_calculus_and_analysis">In calculus and analysis</h2></div>

In <a href="Calculus" title="Calculus">calculus</a>, a function <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"> defined on a <a href="Subset" title="Subset">subset</a> of the <a href="Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> with real values is called monotonic if it is either entirely non-decreasing, or entirely non-increasing.<a href="#cite_note-:1-2">[2]</a> That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.

A function is termed monotonically increasing (also increasing or non-decreasing)<a href="#cite_note-:0-3">[3]</a> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c07a0bc023490be1c08e6c33a9cdc93bec908224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}" loading="lazy"> one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\!\left(x\right)\leq f\!\left(y\right)}">

<mrow>

</mrow>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\!\left(x\right)\leq f\!\left(y\right)}</annotation>

</semantics>

</math><img src="./a6cddeb7f21c061d758a435783ad57163a15c632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.759ex; height:2.843ex;" alt="{\displaystyle f\!\left(x\right)\leq f\!\left(y\right)}" loading="lazy">, 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"> preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing or non-increasing)<a href="#cite_note-:0-3">[3]</a> if, whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c07a0bc023490be1c08e6c33a9cdc93bec908224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\!\left(x\right)\geq f\!\left(y\right)}">

<mrow>

</mrow>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\!\left(x\right)\geq f\!\left(y\right)}</annotation>

</semantics>

</math><img src="./6eb27491d0ae751c3d53fc364a89524623dfba77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.759ex; height:2.843ex;" alt="{\displaystyle f\!\left(x\right)\geq f\!\left(y\right)}" loading="lazy">, so it reverses the order (see Figure 2).

If the order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./440568a09c3bfdf0e1278bfa79eb137c04e94035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }" loading="lazy"> in the definition of monotonicity is replaced by the strict order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./33737c89a17785dacc8638b4d66db3d5c8670de1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}" loading="lazy">, one obtains a stronger requirement. A function with this property is called strictly increasing (also increasing).<a href="#cite_note-:0-3">[3]</a><a href="#cite_note-:2-4">[4]</a> Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing (also decreasing).<a href="#cite_note-:0-3">[3]</a><a href="#cite_note-:2-4">[4]</a> A function with either property is called strictly monotone. Functions that are strictly monotone are <a href="One-to-one_function" class="mw-redirect" title="One-to-one function">one-to-one</a> (because for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> not equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">, either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./aeb239de6fee56ea8b6a65f7858d95b87632069f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x<y}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6e8432c5c4451b66818abae111d41f27d6de8623.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.176ex;" alt="{\displaystyle x>y}" loading="lazy"> and so, by monotonicity, either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\!\left(x\right)<f\!\left(y\right)}">

<mrow>

</mrow>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\!\left(x\right)<f\!\left(y\right)}</annotation>

</semantics>

</math><img src="./7e45fc569a5e9624c01674fc06cb02b3d4cd35de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.759ex; height:2.843ex;" alt="{\displaystyle f\!\left(x\right)<f\!\left(y\right)}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\!\left(x\right)>f\!\left(y\right)}">

<mrow>

</mrow>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\!\left(x\right)>f\!\left(y\right)}</annotation>

</semantics>

</math><img src="./728fae40f016d92409c728fcda0e38bab9b8cd80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.759ex; height:2.843ex;" alt="{\displaystyle f\!\left(x\right)>f\!\left(y\right)}" loading="lazy">, thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\!\left(x\right)\neq f\!\left(y\right)}">

<mrow>

</mrow>

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\!\left(x\right)\neq f\!\left(y\right)}</annotation>

</semantics>

</math><img src="./af8a7e076236ea003e1d23c3b3269d00091f4d9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.759ex; height:2.843ex;" alt="{\displaystyle f\!\left(x\right)\neq f\!\left(y\right)}" loading="lazy">.)

To avoid ambiguity, the terms weakly monotone, weakly increasing and weakly decreasing are often used to refer to non-strict monotonicity.

The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing". For example, the non-monotonic function shown in figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing.

A function <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 said to be absolutely monotonic over an interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a,b\right)}">

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(a,b\right)}</annotation>

</semantics>

</math><img src="./4c254f2a8eaae4e3c7e8f2192c196d621e6c8659.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle \left(a,b\right)}" loading="lazy"> if the derivatives of all orders 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"> are <a href="Nonnegative" class="mw-redirect" title="Nonnegative">nonnegative</a> or all <a href="Nonpositive" class="mw-redirect" title="Nonpositive">nonpositive</a> at all points on the interval.

<div class="mw-heading mw-heading3"><h3 id="Inverse_of_function">Inverse of function</h3></div>

All strictly monotonic functions are <a href="Inverse_function" title="Inverse function">invertible</a> because they are guaranteed to have a one-to-one mapping from their range to their domain.

However, functions that are only weakly monotone are not invertible because they are constant on some interval (and therefore are not one-to-one).

A function may be strictly monotonic over a limited a range of values and thus have an inverse on that range even though it is not strictly monotonic everywhere. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=g(x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./26c08f8fd3471dad5e2c45c2f753ffd7c9aba4ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.509ex; height:2.843ex;" alt="{\displaystyle y=g(x)}" loading="lazy"> is strictly increasing on the range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

</semantics>

</math><img src="./9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy">, then it has an inverse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=h(y)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./04605d887de339e440c9b6a6026f3bc8c53d70ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.732ex; height:2.843ex;" alt="{\displaystyle x=h(y)}" loading="lazy"> on the range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [g(a),g(b)]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [g(a),g(b)]}</annotation>

</semantics>

</math><img src="./091dac07a170589d4ed7e965623e55d374a9de23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.406ex; height:2.843ex;" alt="{\displaystyle [g(a),g(b)]}" loading="lazy">.

The term monotonic is sometimes used in place of strictly monotonic, so a source may state that all monotonic functions are invertible when they really mean that all strictly monotonic functions are invertible.

<div class="mw-heading mw-heading3"><h3 id="Monotonic_transformation">Monotonic transformation</h3></div>

The term monotonic transformation (or monotone transformation) may also cause confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a <a href="Utility_function" class="mw-redirect" title="Utility function">utility function</a> being preserved across a monotonic transform (see also <a href="Monotone_preferences" title="Monotone preferences">monotone preferences</a>).<a href="#cite_note-5">[5]</a> In this context, the term "monotonic transformation" refers to a positive monotonic transformation and is intended to distinguish it from a "negative monotonic transformation," which reverses the order of the numbers.<a href="#cite_note-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Some_basic_applications_and_results">Some basic applications and results</h3></div>

The following properties are true for a monotonic function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b1cacd5f7bbe1027cc75fbe2fbd9cb5e79485302.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }" loading="lazy">:

</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 href="Limit_of_a_function" title="Limit of a function">limits</a> from the right and from the left at every point of its <a href="Domain_of_a_function" title="Domain of a function">domain</a>;</li>

</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 limit at positive or negative infinity (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm \infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c586ae37f8efec026b8a4ea3f6a5253576c2c4e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle \pm \infty }" loading="lazy">) of either a real number, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy">.</li>

</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"> can only have <a href="Jump_discontinuities" class="mw-redirect" title="Jump discontinuities">jump discontinuities</a>;</li>

</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"> can only have <a href="Countably" class="mw-redirect" title="Countably">countably</a> many <a href="Discontinuities_of_monotone_functions" title="Discontinuities of monotone functions">discontinuities</a> in its domain. The discontinuities, however, do not necessarily consist of isolated points and may even be dense in an interval (a, b). For example, for any <a href="Summable_sequence" class="mw-redirect" title="Summable sequence">summable sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(a_{i})">

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</semantics>

</math><img src="./6b5da80cd4cfba3b02a32730b725b17fbad89cc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.839ex; height:2.843ex;" alt="(a_{i})" loading="lazy"> of positive numbers and any enumeration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q_{i})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8c9c8341c4e830b2e3c6213262f4f3ecc59887d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.646ex; height:2.843ex;" alt="{\displaystyle (q_{i})}" loading="lazy"> of the <a href="Rational_number" title="Rational number">rational numbers</a>, the monotonically increasing function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sum _{q_{i}\leq x}a_{i}}">

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\sum _{q_{i}\leq x}a_{i}}</annotation>

</semantics>

</math> is continuous exactly at every irrational number (cf. picture). It is the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of the <a href="Discrete_measure" title="Discrete measure">discrete measure</a> on the rational numbers, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./0bc77764b2e74e64a63341054fa90f3e07db275f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}" loading="lazy"> is the weight of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./2752dcbff884354069fe332b8e51eb0a70a531b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.837ex; height:2.009ex;" alt="{\displaystyle q_{i}}" loading="lazy">.</li>

</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="Differentiable" class="mw-redirect" title="Differentiable">differentiable</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{*}\in {\mathbb {R}}}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{*}\in {\mathbb {R}}}</annotation>

</semantics>

</math><img src="./b27381c729d63c931d5b5d3a06a52bda2e68e222.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.903ex; height:2.343ex;" alt="{\displaystyle x^{*}\in {\mathbb {R}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x^{*})>0}">

<msup>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f'(x^{*})>0}</annotation>

</semantics>

</math><img src="./260fbceb8bc06a13f43fbe16f5058d87b4aa3087.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.459ex; height:3.009ex;" alt="{\displaystyle f'(x^{*})>0}" loading="lazy">, then there is a non-degenerate <a href="Interval_(mathematics)" title="Interval (mathematics)"> interval</a> I such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{*}\in I}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{*}\in I}</annotation>

</semantics>

</math><img src="./195b95e733ef44a2740ced7741bb85c93a96c847.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.396ex; height:2.343ex;" alt="{\displaystyle x^{*}\in I}" loading="lazy"> and <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 increasing on I. As a partial converse, if f is differentiable and increasing on an interval, I, then its derivative is positive at every point in I.</li></ul>

These properties are the reason why monotonic functions are useful in technical work in <a href="Mathematical_analysis" title="Mathematical analysis">analysis</a>. Other important properties of these functions include:

</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 monotonic function defined on an <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy">, then <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="Derivative" title="Derivative">differentiable</a> <a href="Almost_everywhere" title="Almost everywhere">almost everywhere</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy">; i.e. the set of numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> such that <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 in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> has <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue</a> <a href="Measure_zero" class="mw-redirect" title="Measure zero">measure zero</a>. In addition, this result cannot be improved to countable: see <a href="Cantor_function" title="Cantor function">Cantor function</a>.</li>

<li>if this set is countable, then <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 absolutely continuous</li>

</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 monotonic function defined on an interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[a,b\right]}">

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left[a,b\right]}</annotation>

</semantics>

</math><img src="./f30926fb280a9fdf66fd931e14d4363cb824feaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle \left[a,b\right]}" loading="lazy">, then <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_integral" title="Riemann integral">Riemann integrable</a>.</li></ul>

An important application of monotonic functions is in <a href="Probability_theory" title="Probability theory">probability theory</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a <a href="Random_variable" title="Random variable">random variable</a>, its <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}\!\left(x\right)={\text{Prob}}\!\left(X\leq x\right)}">

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X}\!\left(x\right)={\text{Prob}}\!\left(X\leq x\right)}</annotation>

</semantics>

</math><img src="./3c3b7294a8678c1ed20b608eb9e4a6d8267760f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.531ex; height:2.843ex;" alt="{\displaystyle F_{X}\!\left(x\right)={\text{Prob}}\!\left(X\leq x\right)}" loading="lazy"> is a monotonically increasing function.

A function is <a href="Unimodal_function" class="mw-redirect" title="Unimodal function">unimodal</a> if it is monotonically increasing up to some point (the <a href="Mode_(statistics)" title="Mode (statistics)">mode</a>) and then monotonically decreasing.

When <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 strictly monotonic function, then <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="Injective" class="mw-redirect" title="Injective">injective</a> on its domain, and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> is the <a href="Range_of_a_function" title="Range of a function">range</a> 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">, then there is an <a href="Inverse_function" title="Inverse function">inverse function</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> for <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">. In contrast, each constant function is monotonic, but not injective,<a href="#cite_note-7">[7]</a> and hence cannot have an inverse.

The graphic shows six monotonic functions. Their simplest forms are shown in the plot area and the expressions used to create them are shown on the y-axis.

<div class="mw-heading mw-heading2"><h2 id="In_topology">In topology</h2></div>

A map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is said to be monotone if each of its <a href="Fiber_(mathematics)#Fiber_in_naive_set_theory" title="Fiber (mathematics)">fibers</a> is <a href="Connected_(topology)" class="mw-redirect" title="Connected (topology)">connected</a>; that is, for each element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in Y,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y\in Y,}</annotation>

</semantics>

</math><img src="./75e1353f0febe9bcf693d849ec82ce8d94e5f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.416ex; height:2.509ex;" alt="{\displaystyle y\in Y,}" loading="lazy"> the (possibly empty) set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(y)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./8b357745fa4a2178733a502b4432072be8222fd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.618ex; height:3.176ex;" alt="{\displaystyle f^{-1}(y)}" loading="lazy"> is a connected <a href="Subspace_topology" title="Subspace topology">subspace</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="In_functional_analysis">In functional analysis</h2></div>

In <a href="Functional_analysis" title="Functional analysis">functional analysis</a> on a <a href="Topological_vector_space" title="Topological vector space">topological vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, a (possibly non-linear) operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:X\rightarrow X^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T:X\rightarrow X^{*}}</annotation>

</semantics>

</math><img src="./4f5555d4d4e14041de9e01c07a31cd646455e1a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.219ex; height:2.343ex;" alt="{\displaystyle T:X\rightarrow X^{*}}" loading="lazy"> is said to be a monotone operator if

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (Tu-Tv,u-v)\geq 0\quad \forall u,v\in X.}</annotation>

</semantics>

</math>

<a href="Kachurovskii's_theorem" title="Kachurovskii's theorem">Kachurovskii's theorem</a> shows that <a href="Convex_function" title="Convex function">convex functions</a> on <a href="Banach_space" title="Banach space">Banach spaces</a> have monotonic operators as their derivatives.

A subset <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"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times X^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times X^{*}}</annotation>

</semantics>

</math><img src="./aa423f368993d2d9347913501622e3cebc17374f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.871ex; height:2.343ex;" alt="{\displaystyle X\times X^{*}}" loading="lazy"> is said to be a monotone set if for every pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [u_{1},w_{1}]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [u_{1},w_{1}]}</annotation>

</semantics>

</math><img src="./42f2c67bc4887974d491ba4a419dc798ed50d8cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.43ex; height:2.843ex;" alt="{\displaystyle [u_{1},w_{1}]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [u_{2},w_{2}]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [u_{2},w_{2}]}</annotation>

</semantics>

</math><img src="./32202d66739c2039a8b74e861330c713a44db704.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.43ex; height:2.843ex;" alt="{\displaystyle [u_{2},w_{2}]}" loading="lazy"> in <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">,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (w_{1}-w_{2},u_{1}-u_{2})\geq 0.}</annotation>

</semantics>

</math>

</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 said to be maximal monotone if it is maximal among all monotone sets in the sense of set inclusion. The graph of a monotone operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(T)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./140a25dc4428018d67e29b3eb54b07f2bc68d4de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.272ex; height:2.843ex;" alt="{\displaystyle G(T)}" loading="lazy"> is a monotone set. A monotone operator is said to be maximal monotone if its graph is a maximal monotone set.

<div class="mw-heading mw-heading2"><h2 id="In_order_theory">In order theory</h2></div>

Order theory deals with arbitrary <a href="Partially_ordered_set" title="Partially ordered set">partially ordered sets</a> and <a href="Preorder" title="Preorder">preordered sets</a> as a generalization of real numbers. The above definition of monotonicity is relevant in these cases as well. However, the terms "increasing" and "decreasing" are avoided, since their conventional pictorial representation does not apply to orders that are not <a href="Total_order" title="Total order">total</a>. Furthermore, the <a href="Strict_order" class="mw-redirect" title="Strict order">strict</a> relations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./33737c89a17785dacc8638b4d66db3d5c8670de1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle <}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle >}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./1b27b77ab4e3293ea9ce65cef60fea655c398423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle >}" loading="lazy"> are of little use in many non-total orders and hence no additional terminology is introduced for them.

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

</mstyle>

</mrow>

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

</semantics>

</math><img src="./440568a09c3bfdf0e1278bfa79eb137c04e94035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }" loading="lazy"> denote the partial order relation of any partially ordered set, a monotone function, also called isotone, or <style data-mw-deduplicate="TemplateStyles:r1238216509">

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

.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}

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

</style>order-preserving, satisfies the property

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\leq y\implies f(x)\leq f(y)}</annotation>

</semantics>

</math>

for all x and y in its domain. The composite of two monotone mappings is also monotone.

The <a href="Duality_(order_theory)" title="Duality (order theory)">dual</a> notion is often called antitone, anti-monotone, or order-reversing. Hence, an antitone function f satisfies the property

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\leq y\implies f(y)\leq f(x),}</annotation>

</semantics>

</math>

for all x and y in its domain.

A <a href="Constant_function" title="Constant function">constant function</a> is both monotone and antitone; conversely, if f is both monotone and antitone, and if the domain of f is a <a href="Lattice_(order)" title="Lattice (order)">lattice</a>, then f must be constant.

Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are found in these places. Some notable special monotone functions are <a href="Order_embedding" title="Order embedding">order embeddings</a> (functions for which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c07a0bc023490be1c08e6c33a9cdc93bec908224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.343ex;" alt="{\displaystyle x\leq y}" loading="lazy"> <a href="If_and_only_if" title="If and only if">if and only if</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\leq f(y))}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ca2b65c72e2f561be6f1f2cfdb5120f9b07b98da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.664ex; height:2.843ex;" alt="{\displaystyle f(x)\leq f(y))}" loading="lazy"> and <a href="Order_isomorphism" title="Order isomorphism">order isomorphisms</a> (<a href="Surjective" class="mw-redirect" title="Surjective">surjective</a> order embeddings).

<div class="mw-heading mw-heading2"><h2 id="In_the_context_of_search_algorithms">In the context of search algorithms</h2></div>

In the context of <a href="Search_algorithm" title="Search algorithm">search algorithms</a> monotonicity (also called consistency) is a condition applied to <a href="Heuristic_function" class="mw-redirect" title="Heuristic function">heuristic functions</a>. A heuristic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(n)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./125976c5970a395422e1baf572a26f31ae5e0b7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.543ex; height:2.843ex;" alt="{\displaystyle h(n)}" loading="lazy"> is monotonic if, for every node n and every successor n' of n generated by any action a, the estimated cost of reaching the goal from n is no greater than the step cost of getting to n' plus the estimated cost of reaching the goal from n',

<mrow>

<mrow>

<msup>

</msup>

</mrow>

</mrow>

<mrow>

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(n)\leq c\left(n,a,n'\right)+h\left(n'\right).}</annotation>

</semantics>

</math>

This is a form of <a href="Triangle_inequality" title="Triangle inequality">triangle inequality</a>, with n, n', and the goal Gn closest to n. Because every monotonic heuristic is also <a href="Admissible_heuristic" title="Admissible heuristic">admissible</a>, monotonicity is a stricter requirement than admissibility. Some <a href="Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristic algorithms</a> such as <a href="A*_search_algorithm" title="A* search algorithm">A*</a> can be proven <a href="Asymptotically_optimal_algorithm" title="Asymptotically optimal algorithm">optimal</a> provided that the heuristic they use is monotonic.<a href="#cite_note-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="In_Boolean_functions">In Boolean functions</h2></div>

<td>

</td></tr></tbody></table>

<td>

</td></tr></tbody></table>

In <a href="Boolean_algebra" title="Boolean algebra">Boolean algebra</a>, a monotonic function is one such that for all ai and bi in {0,1}, if a1 ≤ b1, a2 ≤ b2, ..., an ≤ bn (i.e. the Cartesian product {0, 1}n is ordered <a href="Coordinatewise_order" class="mw-redirect" title="Coordinatewise order">coordinatewise</a>), then f(a1, ..., an) ≤ f(b1, ..., bn). In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. Graphically, this means that an n-ary Boolean function is monotonic when its representation as an <a href="Hypercube" title="Hypercube">n-cube</a> labelled with truth values has no upward edge from true to false. (This labelled <a href="Hasse_diagram" title="Hasse diagram">Hasse diagram</a> is the <a href="Duality_(mathematics)#Dimension-reversing_dualities" title="Duality (mathematics)">dual</a> of the function's labelled <a href="Venn_diagram" title="Venn diagram">Venn diagram</a>, which is the more common representation for n ≤ 3.)

The monotonic Boolean functions are precisely those that can be defined by an expression combining the inputs (which may appear more than once) using only the operators <a href="Logical_conjunction" title="Logical conjunction">and</a> and <a href="Logical_disjunction" title="Logical disjunction">or</a> (in particular <a href="Negation" title="Negation">not</a> is forbidden). For instance "at least two of a, b, c hold" (the ternary <a href="Majority_function" title="Majority function">majority function</a>) is a monotonic function of a, b, c, since it can be written for instance as ((a and b) or (a and c) or (b and c)).

The number of such functions on n variables is known as the <a href="Dedekind_number" title="Dedekind number">Dedekind number</a> of n.

<a href="SAT_solving" class="mw-redirect" title="SAT solving">SAT solving</a>, generally an <a href="NP-hard" class="mw-redirect" title="NP-hard">NP-hard</a> task, can be achieved efficiently when all involved functions and predicates are monotonic and Boolean.<a href="#cite_note-9">[9]</a>

<ul><li><a href="Monotone_cubic_interpolation" title="Monotone cubic interpolation">Monotone cubic interpolation</a></li>

<li><a href="Pseudo-monotone_operator" title="Pseudo-monotone operator">Pseudo-monotone operator</a></li>

<li><a href="Spearman's_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's rank correlation coefficient</a> - measure of monotonicity in a set of data</li>

<li><a href="Total_monotonicity" class="mw-redirect" title="Total monotonicity">Total monotonicity</a></li>

<li><a href="Cyclical_monotonicity" title="Cyclical monotonicity">Cyclical monotonicity</a></li>

<li><a href="Operator_monotone_function" title="Operator monotone function">Operator monotone function</a></li>

<li><a href="Monotone_set_function" class="mw-redirect" title="Monotone set function">Monotone set function</a></li>

<li><a href="Absolutely_and_completely_monotonic_functions_and_sequences" title="Absolutely and completely monotonic functions and sequences">Absolutely and completely monotonic functions and sequences</a></li></ul>

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

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

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

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

</style><div class="reflist">

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

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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

</style><cite id="CITEREFClaphamNicholson2014" class="citation book cs1">Clapham, Christopher; Nicholson, James (2014). Oxford Concise Dictionary of Mathematics (5th ed.). Oxford University Press.</cite>

</li>

<li id="cite_note-:1-2">^ <a href="#cite_ref-:1_2-0">a</a> <a href="#cite_ref-:1_2-1">b</a> <cite id="CITEREFStover" class="citation web cs1">Stover, Christopher. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/MonotonicFunction.html">"Monotonic Function"</a>. Wolfram MathWorld. Retrieved 2018-01-29.</cite>

</li>

<li id="cite_note-:0-3">^ <a href="#cite_ref-:0_3-0">a</a> <a href="#cite_ref-:0_3-1">b</a> <a href="#cite_ref-:0_3-2">c</a> <a href="#cite_ref-:0_3-3">d</a> <a href="#cite_ref-:0_3-4">e</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Monotone_function">"Monotone function"</a>. Encyclopedia of Mathematics. Retrieved 2018-01-29.</cite>

</li>

<li id="cite_note-:2-4">^ <a href="#cite_ref-:2_4-0">a</a> <a href="#cite_ref-:2_4-1">b</a> <cite id="CITEREFSpivak1994" class="citation book cs1">Spivak, Michael (1994). Calculus. Houston, Texas: Publish or Perish, Inc. p. 192. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-914098-89-6</bdi>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> See the section on Cardinal Versus Ordinal Utility in <a href="#CITEREFSimonBlume1994">Simon & Blume (1994)</a>.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFVarian2010" class="citation book cs1">Varian, Hal R. (2010). Intermediate Microeconomics (8th ed.). W. W. Norton & Company. p. 56. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780393934243</bdi>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> if its domain has more than one element

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> Conditions for optimality: Admissibility and consistency pg. 94–95 (<a href="#CITEREFRussellNorvig2010">Russell & Norvig 2010</a>).

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFBaylessBaylessHoosHu2015" class="citation conference cs1">Bayless, Sam; Bayless, Noah; Hoos, Holger H.; Hu, Alan J. (2015). <a rel="nofollow" class="external text" href="https://ojs.aaai.org/index.php/AAAI/article/view/9755">SAT Modulo Monotonic Theories</a>. Proc. 29th AAAI Conf. on Artificial Intelligence. AAAI Press. pp. 3702–3709. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1406.0043">1406.0043</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1609%2Faaai.v29i1.9755">10.1609/aaai.v29i1.9755</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20231211023402/https://ojs.aaai.org/index.php/AAAI/article/view/9755">Archived</a> from the original on Dec 11, 2023.</cite>

</li>

</ol></div></div>

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

<ul><li><cite id="CITEREFBartle1976" class="citation book cs1">Bartle, Robert G. (1976). The elements of real analysis (second ed.).</cite></li>

<li><cite id="CITEREFGrätzer1971" class="citation book cs1">Grätzer, George (1971). Lattice theory: first concepts and distributive lattices. W. H. Freeman. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7167-0442-0</bdi>.</cite></li>

<li><cite id="CITEREFPembertonRau,_Nicholas2001" class="citation book cs1">Pemberton, Malcolm; Rau, Nicholas (2001). Mathematics for economists: an introductory textbook. Manchester University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7190-3341-1</bdi>.</cite></li>

<li><cite id="CITEREFRenardy,_MichaelRogers,_Robert_C.2004" class="citation book cs1">Renardy, Michael & Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. p. 356. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-00444-0</bdi>.</cite></li>

<li><cite id="CITEREFRiesz,_FrigyesBéla_Szőkefalvi-Nagy1990" class="citation book cs1">Riesz, Frigyes & Béla Szőkefalvi-Nagy (1990). Functional Analysis. Courier Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-66289-3</bdi>.</cite></li>

<li><cite id="CITEREFRussellNorvig2010" class="citation book cs1">Russell, Stuart J.; Norvig, Peter (2010). Artificial Intelligence: A Modern Approach (3rd ed.). Upper Saddle River, New Jersey: Prentice Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-604259-4</bdi>.</cite></li>

<li><cite id="CITEREFSimonBlume1994" class="citation book cs1">Simon, Carl P.; Blume, Lawrence (April 1994). Mathematics for Economists (first ed.). Norton. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-393-95733-4</bdi>.</cite> (Definition 9.31)</li></ul>

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

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Monotone_function">"Monotone 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><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/ConvergenceOfAMonotonicSequence/">Convergence of a Monotonic Sequence</a> by Anik Debnath and Thomas Roxlo (The Harker School), <a href="Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>.</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/MonotonicFunction.html">"Monotonic Function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li></ul>

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

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

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

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

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

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

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

</style></div><div role="navigation" class="navbox" aria-labelledby="Order_theory153" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r1239400231">

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

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

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

</style><div id="Order_theory153" style="font-size:114%;margin:0 4em"><a href="Order_theory" title="Order theory">Order theory</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="List_of_order_theory_topics" title="List of order theory topics">Topics</a></li>

<li><a href="Glossary_of_order_theory" title="Glossary of order theory">Glossary</a></li>

<li>Category</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Binary_relation" title="Binary relation">Binary relation</a></li>

<li><a href="Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></li>

<li><a href="Cyclic_order" title="Cyclic order">Cyclic order</a></li>

<li><a href="Lattice_(order)" title="Lattice (order)">Lattice</a></li>

<li><a href="Partially_ordered_set" title="Partially ordered set">Partial order</a></li>

<li><a href="Preorder" title="Preorder">Preorder</a></li>

<li><a href="Total_order" title="Total order">Total order</a></li>

<li><a href="Weak_ordering" title="Weak ordering">Weak ordering</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Boolean_prime_ideal_theorem" title="Boolean prime ideal theorem">Boolean prime ideal theorem</a></li>

<li><a href="Cantor%E2%80%93Bernstein_theorem" title="Cantor–Bernstein theorem">Cantor–Bernstein theorem</a></li>

<li><a href="Cantor's_isomorphism_theorem" title="Cantor's isomorphism theorem">Cantor's isomorphism theorem</a></li>

<li><a href="Dilworth's_theorem" title="Dilworth's theorem">Dilworth's theorem</a></li>

<li><a href="Dushnik%E2%80%93Miller_theorem" title="Dushnik–Miller theorem">Dushnik–Miller theorem</a></li>

<li><a href="Hausdorff_maximal_principle" title="Hausdorff maximal principle">Hausdorff maximal principle</a></li>

<li><a href="Knaster%E2%80%93Tarski_theorem" title="Knaster–Tarski theorem">Knaster–Tarski theorem</a></li>

<li><a href="Kruskal's_tree_theorem" title="Kruskal's tree theorem">Kruskal's tree theorem</a></li>

<li><a href="Laver's_theorem" title="Laver's theorem">Laver's theorem</a></li>

<li><a href="Mirsky's_theorem" title="Mirsky's theorem">Mirsky's theorem</a></li>

<li><a href="Szpilrajn_extension_theorem" title="Szpilrajn extension theorem">Szpilrajn extension theorem</a></li>

<li><a href="Zorn's_lemma" title="Zorn's lemma">Zorn's lemma</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties & Types (<a href="List_of_order_structures_in_mathematics" title="List of order structures in mathematics">list</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric</a></li>

<li><a href="Asymmetric_relation" title="Asymmetric relation">Asymmetric</a></li>

<li><a href="Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a>

<ul><li><a href="List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">topics</a></li></ul></li>

<li><a href="Completeness_(order_theory)" title="Completeness (order theory)">Completeness</a></li>

<li><a href="Connected_relation" title="Connected relation">Connected</a></li>

<li><a href="Covering_relation" title="Covering relation">Covering</a></li>

<li><a href="Dense_order" title="Dense order">Dense</a></li>

<li><a href="Directed_set" title="Directed set">Directed</a></li>

<li>(<a href="Partial_equivalence_relation" title="Partial equivalence relation">Partial</a>) <a href="Equivalence_relation" title="Equivalence relation">Equivalence</a></li>

<li><a href="Foundational_relation" class="mw-redirect" title="Foundational relation">Foundational</a></li>

<li><a href="Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li>

<li><a href="Homogeneous_relation" title="Homogeneous relation">Homogeneous</a></li>

<li><a href="Idempotent_relation" title="Idempotent relation">Idempotent</a></li>

<li><a href="Lattice_(order)" title="Lattice (order)">Lattice</a>

<ul><li><a href="Bounded_lattice" class="mw-redirect" title="Bounded lattice">Bounded</a></li>

<li><a href="Complemented_lattice" title="Complemented lattice">Complemented</a></li>

<li><a href="Complete_lattice" title="Complete lattice">Complete</a></li>

<li><a href="Distributive_lattice" title="Distributive lattice">Distributive</a></li>

<li><a href="Reflexive_relation" title="Reflexive relation">Reflexive</a></li>

<li><a href="Partial_order" class="mw-redirect" title="Partial order">Partial order</a>

<ul><li><a href="Chain-complete_partial_order" class="mw-redirect" title="Chain-complete partial order">Chain-complete</a></li>

<li><a href="Graded_poset" title="Graded poset">Graded</a></li>

<li><a href="Eulerian_poset" title="Eulerian poset">Eulerian</a></li>

<li><a href="Strict_partial_order" class="mw-redirect" title="Strict partial order">Strict</a></li></ul></li>

<li><a href="Prefix_order" title="Prefix order">Prefix order</a></li>

<li><a href="Preorder" title="Preorder">Preorder</a>

<ul><li><a href="Total_preorder" class="mw-redirect" title="Total preorder">Total</a></li></ul></li>

<li><a href="Semilattice" title="Semilattice">Semilattice</a></li>

<li><a href="Semiorder" title="Semiorder">Semiorder</a></li>

<li><a href="Symmetric_relation" title="Symmetric relation">Symmetric</a></li>

<li><a href="Total_relation" title="Total relation">Total</a></li>

<li><a href="Tolerance_relation" title="Tolerance relation">Tolerance</a></li>

<li><a href="Transitive_relation" title="Transitive relation">Transitive</a></li>

<li><a href="Well-founded_relation" title="Well-founded relation">Well-founded</a></li>

<li><a href="Well-quasi-ordering" title="Well-quasi-ordering">Well-quasi-ordering</a> (<a href="Better-quasi-ordering" title="Better-quasi-ordering">Better</a>)</li>

<li>(<a href="Prewellordering" title="Prewellordering">Pre</a>) <a href="Well-order" title="Well-order">Well-order</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Composition_of_relations" title="Composition of relations">Composition</a></li>

<li><a href="Converse_relation" title="Converse relation">Converse/Transpose</a></li>

<li><a href="Lexicographic_order" title="Lexicographic order">Lexicographic order</a></li>

<li><a href="Linear_extension" title="Linear extension">Linear extension</a></li>

<li><a href="Product_order" title="Product order">Product order</a></li>

<li><a href="Reflexive_closure" title="Reflexive closure">Reflexive closure</a></li>

<li><a href="Series-parallel_partial_order" title="Series-parallel partial order">Series-parallel partial order</a></li>

<li><a href="Star_product" title="Star product">Star product</a></li>

<li><a href="Symmetric_closure" title="Symmetric closure">Symmetric closure</a></li>

<li><a href="Transitive_closure" title="Transitive closure">Transitive closure</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Topology" title="Topology">Topology</a> & Orders</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alexandrov_topology" title="Alexandrov topology">Alexandrov topology</a> & <a href="Specialization_(pre)order" title="Specialization (pre)order">Specialization preorder</a></li>

<li><a href="Ordered_topological_vector_space" title="Ordered topological vector space">Ordered topological vector space</a>

<ul><li><a href="Normal_cone_(functional_analysis)" title="Normal cone (functional analysis)">Normal cone</a></li>

<li><a href="Order_topology_(functional_analysis)" title="Order topology (functional analysis)">Order topology</a></li></ul></li>

<li><a href="Order_topology" title="Order topology">Order topology</a></li>

<li><a href="Topological_vector_lattice" title="Topological vector lattice">Topological vector lattice</a>

<ul><li><a href="Banach_lattice" title="Banach lattice">Banach</a></li>

<li><a href="Fr%C3%A9chet_lattice" title="Fréchet lattice">Fréchet</a></li>

<li><a href="Locally_convex_vector_lattice" title="Locally convex vector lattice">Locally convex</a></li>

<li><a href="Normed_lattice" class="mw-redirect" title="Normed lattice">Normed</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Antichain" title="Antichain">Antichain</a></li>

<li><a href="Cofinal_(mathematics)" title="Cofinal (mathematics)">Cofinal</a></li>

<li><a href="Cofinality" title="Cofinality">Cofinality</a></li>

<li><a href="Comparability" title="Comparability">Comparability</a>

<ul><li><a href="Comparability_graph" title="Comparability graph">Graph</a></li></ul></li>

<li><a href="Duality_(order_theory)" title="Duality (order theory)">Duality</a></li>

<li><a href="Filter_(mathematics)" title="Filter (mathematics)">Filter</a></li>

<li><a href="Hasse_diagram" title="Hasse diagram">Hasse diagram</a></li>

<li><a href="Ideal_(order_theory)" title="Ideal (order theory)">Ideal</a></li>

<ul><li><a href="Subnet_(mathematics)" title="Subnet (mathematics)">Subnet</a></li></ul></li>

<li>

<ul><li><a href="Order_embedding" title="Order embedding">Embedding</a></li>

<li><a href="Order_isomorphism" title="Order isomorphism">Isomorphism</a></li></ul></li>

<li><a href="Order_type" title="Order type">Order type</a></li>

<li><a href="Ordered_field" title="Ordered field">Ordered field</a>

<ul><li><a href="Positive_cone_of_an_ordered_field" class="mw-redirect" title="Positive cone of an ordered field">Positive cone of an ordered field</a></li></ul></li>

<li><a href="Ordered_vector_space" title="Ordered vector space">Ordered vector space</a>

<ul><li><a href="Partially_ordered_space" title="Partially ordered space">Partially ordered</a></li>

<li><a href="Positive_cone_of_an_ordered_vector_space" class="mw-redirect" title="Positive cone of an ordered vector space">Positive cone of an ordered vector space</a></li>

<li><a href="Riesz_space" title="Riesz space">Riesz space</a></li></ul></li>

<li><a href="Partially_ordered_group" title="Partially ordered group">Partially ordered group</a>

<ul><li><a href="Positive_cone_of_a_partially_ordered_group" class="mw-redirect" title="Positive cone of a partially ordered group">Positive cone of a partially ordered group</a></li></ul></li>

<li><a href="Upper_set" title="Upper set">Upper set</a></li>

<li><a href="Young's_lattice" title="Young's lattice">Young's lattice</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-02" href="https://en.wikipedia.org/wiki/?title=Monotonic_function&oldid=1298392473">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.

</div>

</div></div>

</div>

</main>

</div>

</body></html>