Singular function

<h2 id="when-referring-to-functions-with-a-singularity">When referring to functions with a singularity</h2>
<p>When discussing <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a> in general, or more specifically <a href="/facts/Real_analysis/12WjlXCW">real analysis</a> or <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a> or <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a>, it is common for a function which contains a <a href="/facts/Mathematical_singularity/Y5sMQY4I">mathematical singularity</a> to be referred to as a 'singular function'. This is especially true when referring to functions which diverge to infinity at a point or on a boundary. For example, one might say, "<i>1/x</i> becomes singular at the origin, so <i>1/x</i> is a singular function."
</p><p>Advanced techniques for working with functions that contain singularities have been developed in the subject called <a href="/facts/Distribution_(mathematics)/yfaJ4mUp">distributional</a> or <a href="/facts/Generalized_function/fCM4VsyT">generalized function</a> analysis. A <a href="/facts/Weak_derivative/oehSmSMg">weak derivative</a> is defined that allows singular functions to be used in <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equations</a>, etc.
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<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Absolute_continuity/NPKDt9Qn">Absolute continuity</a></li>
<li><a href="/facts/Mathematical_singularity/Y5sMQY4I">Mathematical singularity</a></li>
<li><a href="/facts/Generalized_function/fCM4VsyT">Generalized function</a></li>
<li><a href="/facts/Distribution_(mathematics)/yfaJ4mUp">Distribution</a></li>
<li><a href="/facts/Minkowski%27s_question-mark_function/gum3I0pX">Minkowski's question-mark function</a></li></ul>

<p>(**) This condition depends on the <a href="/facts/References/pLmVxBjJ">references</a>
<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>

<ul><li>Lebesgue, H. (1955–1961), <i>Theory of functions of a real variable</i>, <a href="/facts/F._Ungar/UWsJpwlH">F. Ungar</a></li>
<li>Halmos, P.R. (1950), <i>Measure theory</i>, v. Nostrand</li>
<li>Royden, H.L (1988), <i>Real Analysis</i>, Prentice-Hall, Englewood Cliffs, New Jersey</li>
<li>Lebesgue, H. (1928), <i>Leçons sur l'intégration et la récherche des fonctions primitives</i>, Gauthier-Villars</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>"Singular function", Encyclopedia of Mathematics, EMS Press, 2001 [1994] <a href="https://www.encyclopediaofmath.org/index.php?title=Singular_function" target="_blank">https://www.encyclopediaofmath.org/index.php?title=Singular_function</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Singular function open-in-new

Singular function