SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Microcontinuity</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Microcontinuity</span></span>

</h1>

</header>

<p>In <a href="Nonstandard_analysis" title="Nonstandard analysis">nonstandard analysis</a>, a discipline within classical mathematics, <b>microcontinuity</b> (or <i>S</i>-continuity) of an <a href="Internal_function" class="mw-redirect" title="Internal function">internal function</a> <i>f</i> at a point <i>a</i> is defined as follows:

</p>

<dl><dd>for all <i>x</i> infinitely close to <i>a</i>, the value <i>f</i>(<i>x</i>) is infinitely close to <i>f</i>(<i>a</i>).</dd></dl>

<p>Here <i>x</i> runs through the domain of <i>f</i>. In formulas, this can be expressed as follows:

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./68f8ec36c14cb80f2e78e083d23add919ddfdaa3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x\approx a}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\approx f(a)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./deb8b324a1dad075a6639ef9d8e50d50f317c171.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.834ex; height:2.843ex;" alt="{\displaystyle f(x)\approx f(a)}" loading="lazy"></span>.</dd></dl>

<p>For a function <i>f</i> defined on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.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 {R} }" loading="lazy"></span>, the definition can be expressed in terms of the <a href="Halo_(mathematics)" class="mw-redirect" title="Halo (mathematics)">halo</a> as follows: <i>f</i> is microcontinuous at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d47ef490c028656282fd8b18c44c4939bbfff750.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.526ex; height:2.176ex;" alt="{\displaystyle c\in \mathbb {R} }" loading="lazy"></span> if and only if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(hal(c))\subseteq hal(f(c))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(hal(c))\subseteq hal(f(c))}</annotation>

</semantics>

</math></span><img src="./74253a8e2b34ccfc645cbb2d8434ccd682d92786.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.431ex; height:2.843ex;" alt="{\displaystyle f(hal(c))\subseteq hal(f(c))}" loading="lazy"></span>, where the natural extension of <i>f</i> to the <a href="Hyperreal_number" title="Hyperreal number">hyperreals</a> is still denoted <i>f</i>. Alternatively, the property of microcontinuity at <i>c</i> can be expressed by stating that the composition <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{st}}\circ f}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./538e6c5c4e182b665bdaf369f5619fb606943b2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.294ex; height:2.509ex;" alt="{\displaystyle {\text{st}}\circ f}" loading="lazy"></span> is constant on the halo of <i>c</i>, where "st" is the <a href="Standard_part_function" title="Standard part function">standard part function</a>.

</p>

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

<p>The modern property of continuity of a function was first defined by Bolzano in 1817. However, Bolzano's work was not noticed by the larger mathematical community until its rediscovery in Heine in the 1860s. Meanwhile, <a href="Cauchy" class="mw-redirect" title="Cauchy">Cauchy</a>'s textbook <a href="Cours_d'Analyse" class="mw-redirect" title="Cours d'Analyse">Cours d'Analyse</a> defined continuity in 1821 using <a href="Infinitesimal" title="Infinitesimal">infinitesimals</a> as above.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Continuity_and_uniform_continuity">Continuity and uniform continuity</h2></div>

<p>The property of microcontinuity is typically applied to the natural extension <i>f*</i> of a real function <i>f</i>. Thus, <i>f</i> defined on a real interval <i>I</i> is <a href="Continuous_function" title="Continuous function">continuous</a> if and only if <i>f*</i> is microcontinuous at every point of <i>I</i>. Meanwhile, <i>f</i> is <a href="Uniformly_continuous" class="mw-redirect" title="Uniformly continuous">uniformly continuous</a> on <i>I</i> if and only if <i>f*</i> is microcontinuous at every point (standard and nonstandard) of the natural extension <i>I*</i> of its domain <i>I</i> (see Davis, 1977, p. 96).

</p>

<div class="mw-heading mw-heading2"><h2 id="Example_1">Example 1</h2></div>

<p>The real function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\tfrac {1}{x}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./86ad8bb8ae44173f9fc433075758a1750ecf5c51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.292ex; height:3.343ex;" alt="{\displaystyle f(x)={\tfrac {1}{x}}}" loading="lazy"></span> on the open interval (0,1) is not uniformly continuous because the natural extension <i>f*</i> of <i>f</i> fails to be microcontinuous at an <a href="Infinitesimal" title="Infinitesimal">infinitesimal</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1f34a80ea013edb56e340b19550430a8b6dfd7b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>0}" loading="lazy"></span>. Indeed, for such an <i>a</i>, the values <i>a</i> and <i>2a</i> are infinitely close, but the values of <i>f*</i>, namely <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{a}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b01639881379f5c524b620c7bfa4f6d2e2a089ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.706ex; height:3.343ex;" alt="{\displaystyle {\tfrac {1}{a}}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2a}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2a}}}</annotation>

</semantics>

</math></span><img src="./002715926e86ea3b7011bf1a43cfcebedd5bac08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.528ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2a}}}" loading="lazy"></span> are not infinitely close.

</p>

<div class="mw-heading mw-heading2"><h2 id="Example_2">Example 2</h2></div>

<p>The function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=x^{2}}</annotation>

</semantics>

</math></span><img src="./84ddac4ae10b1aa4a11741c79771a583419fb1fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.9ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{2}}" loading="lazy"></span> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.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 {R} }" loading="lazy"></span> is not uniformly continuous because <i>f*</i> fails to be microcontinuous at an infinite point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\in \mathbb {R} ^{*}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a492697160190bdf6e11996029748959716debf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.637ex; height:2.343ex;" alt="{\displaystyle H\in \mathbb {R} ^{*}}" loading="lazy"></span>. Namely, setting <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e={\tfrac {1}{H}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./6d304549fe6aa8b6e6fa369aeb2957381918a1ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.477ex; height:3.509ex;" alt="{\displaystyle e={\tfrac {1}{H}}}" loading="lazy"></span> and <i>K</i> = <i>H</i> + <i>e</i>, one easily sees that <i>H</i> and <i>K</i> are infinitely close but <i>f</i>*(<i>H</i>) and <i>f</i>*(<i>K</i>) are not infinitely close.

</p>

<div class="mw-heading mw-heading2"><h2 id="Uniform_convergence">Uniform convergence</h2></div>

<p><a href="Uniform_convergence" title="Uniform convergence">Uniform convergence</a> similarly admits a simplified definition in a hyperreal setting. Thus, a sequence <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="./b2702450f0458a5e01a698e248af552a7fab2b50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.509ex;" alt="{\displaystyle f_{n}}" loading="lazy"></span> converges to <i>f</i> uniformly if for all <i>x</i> in the domain of <i>f*</i> and all infinite <i>n</i>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}^{*}(x)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./5f3512b1338c39b83752a58ede045cf60cb50c78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.514ex; height:2.843ex;" alt="{\displaystyle f_{n}^{*}(x)}" loading="lazy"></span> is infinitely close to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}(x)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b6b4088f786acb68c65dda1da1bfd5b022023c3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.514ex; height:2.843ex;" alt="{\displaystyle f^{*}(x)}" loading="lazy"></span>.

</p>

<ul><li><a href="Standard_part_function" title="Standard part function">Standard part function</a></li></ul>

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

<ul><li><a href="Martin_Davis_(mathematician)" title="Martin Davis (mathematician)">Martin Davis</a> (1977) Applied nonstandard analysis. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney. xii+181 pp. <style data-mw-deduplicate="TemplateStyles:r1238218222">

/* 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><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-19897-8</bdi></li>

<li>Gordon, E. I.; Kusraev, A. G.; <a href="Semen_Samsonovich_Kutateladze" class="mw-redirect" title="Semen Samsonovich Kutateladze">Kutateladze</a>, S. S.: Infinitesimal analysis. Updated and revised translation of the 2001 Russian original. Translated by Kutateladze. Mathematics and its Applications, 544. Kluwer Academic Publishers, Dordrecht, 2002.</li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite id="CITEREFBorovikKatz2011" class="citation cs2"><a href="Alexandre_Borovik" title="Alexandre Borovik">Borovik, Alexandre</a>; <a href="Mikhail_Katz" title="Mikhail Katz">Katz, Mikhail G.</a> (2011), "Who gave you the Cauchy--Weierstrass tale? The dual history of rigorous calculus", <i><a href="Foundations_of_Science" title="Foundations of Science">Foundations of Science</a></i>, <b>17</b> (3): <span class="nowrap">245–</span>276, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1108.2885">1108.2885</a></span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10699-011-9235-x">10.1007/s10699-011-9235-x</a></cite>.</span>

</li>

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

/* 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="Infinitesimals97" style="padding:3px"><table class="nowraplinks mw-collapsible expanded navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><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="Infinitesimals97" style="font-size:114%;margin:0 4em"><a href="Infinitesimal" title="Infinitesimal">Infinitesimals</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">History</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

<li><a href="Leibniz's_notation" title="Leibniz's notation">Leibniz's notation</a></li>

<li><a href="Integral_symbol" title="Integral symbol">Integral symbol</a></li>

<li><a href="Criticism_of_nonstandard_analysis" title="Criticism of nonstandard analysis">Criticism of nonstandard analysis</a></li>

<li><i><a href="The_Analyst" title="The Analyst">The Analyst</a></i></li>

<li><i><a href="The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li>

<li><a href="Cavalieri's_principle" title="Cavalieri's principle">Cavalieri's principle</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span class="skin-invert-image" typeof="mw:File"></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related branches</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li>

<li><a href="Nonstandard_calculus" title="Nonstandard calculus">Nonstandard calculus</a></li>

<li><a href="Internal_set_theory" title="Internal set theory">Internal set theory</a></li>

<li><a href="Synthetic_differential_geometry" title="Synthetic differential geometry">Synthetic differential geometry</a></li>

<li><a href="Smooth_infinitesimal_analysis" title="Smooth infinitesimal analysis">Smooth infinitesimal analysis</a></li>

<li><a href="Constructive_nonstandard_analysis" title="Constructive nonstandard analysis">Constructive nonstandard analysis</a></li>

<li><a href="Infinitesimal_strain_theory" title="Infinitesimal strain theory">Infinitesimal strain theory (physics)</a></li></ul>

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

<ul><li><a href="Differential_(mathematics)" title="Differential (mathematics)">Differentials</a></li>

<li><a href="Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li>

<li><a href="Dual_number" title="Dual number">Dual numbers</a></li>

<li><a href="Surreal_number" title="Surreal number">Surreal numbers</a></li></ul>

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

<ul><li><a href="Standard_part_function" title="Standard part function">Standard part function</a></li>

<li><a href="Transfer_principle" title="Transfer principle">Transfer principle</a></li>

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

<li><a href="Increment_theorem" title="Increment theorem">Increment theorem</a></li>

<li><a href="Monad_(nonstandard_analysis)" title="Monad (nonstandard analysis)">Monad</a></li>

<li><a href="Internal_set" title="Internal set">Internal set</a></li>

<li><a href="Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li>

<li><a href="Hyperfinite_set" title="Hyperfinite set">Hyperfinite set</a></li>

<li><a href="Law_of_continuity" title="Law of continuity">Law of continuity</a></li>

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

<li><a href="Transcendental_law_of_homogeneity" title="Transcendental law of homogeneity">Transcendental law of homogeneity</a></li></ul>

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

<ul><li><a href="Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li>

<li><a href="Abraham_Robinson" title="Abraham Robinson">Abraham Robinson</a></li>

<li><a href="Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a></li>

<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li></ul>

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

<ul><li><a href="Analyse_des_Infiniment_Petits_pour_l'Intelligence_des_Lignes_Courbes" class="mw-redirect" title="Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes">Analyse des Infiniment Petits</a></li>

<li><a href="Elementary_Calculus%3A_An_Infinitesimal_Approach" title="Elementary Calculus: An Infinitesimal Approach">Elementary Calculus</a></li>

<li><a href="Cours_d'analyse" title="Cours d'analyse">Cours d'analyse</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 2024-12-03" href="https://en.wikipedia.org/wiki/?title=Microcontinuity&oldid=1260890707">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>