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Difference between revisions of "Discontinuous function"

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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033000/d0330001.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033000/d0330002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033000/d0330003.png" /> are topological spaces, that is not a [[Continuous function|continuous function]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033000/d0330004.png" />. The [[Baire classes|Baire classes]], the piecewise-continuous functions and the step functions are important classes of discontinuous real-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033000/d0330005.png" />.
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A function $f : X \rightarrow Y$, where $X$ and $Y$ are topological spaces, that is not a [[continuous function]] on $X$. The [[Baire classes]], the piecewise-continuous functions and the [[step function]]s are important classes of discontinuous real-valued functions $f : X \rightarrow \mathbf{R}$.
  
Discontinuous functions occur, for example, when integrating elementary functions with respect to a parameter (see [[Dirichlet discontinuous multiplier|Dirichlet discontinuous multiplier]]), when calculating the sum of a series in which the terms are elementary functions, in particular when calculating the sum of a trigonometric series, and in optimal control problems.
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Discontinuous functions occur, for example, when integrating elementary functions with respect to a parameter (see [[Dirichlet discontinuous multiplier]]), when calculating the sum of a series in which the terms are elementary functions, in particular when calculating the sum of a trigonometric series, and in optimal control problems.
  
 
Examples.
 
Examples.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033000/d0330006.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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\sum_{n=0}^\infty \frac{x^2}{(1+x^2)^n} = \begin{cases} 0 & \text{if}\ x = 0 \ , \\ 1+x^2 & \text{otherwise} \ . \end{cases}
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\sum_{n=1}^\infty \frac{\sin nx}{n} = \begin{cases} 0 & \text{if}\ x = 0 \ , \\ \frac{\pi-x}{2} & \text{if} \ 0 < x < \pi \ . \end{cases}
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033000/d0330007.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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Latest revision as of 17:19, 20 November 2016

A function $f : X \rightarrow Y$, where $X$ and $Y$ are topological spaces, that is not a continuous function on $X$. The Baire classes, the piecewise-continuous functions and the step functions are important classes of discontinuous real-valued functions $f : X \rightarrow \mathbf{R}$.

Discontinuous functions occur, for example, when integrating elementary functions with respect to a parameter (see Dirichlet discontinuous multiplier), when calculating the sum of a series in which the terms are elementary functions, in particular when calculating the sum of a trigonometric series, and in optimal control problems.

Examples.

$$ \sum_{n=0}^\infty \frac{x^2}{(1+x^2)^n} = \begin{cases} 0 & \text{if}\ x = 0 \ , \\ 1+x^2 & \text{otherwise} \ . \end{cases} $$ $$ \sum_{n=1}^\infty \frac{\sin nx}{n} = \begin{cases} 0 & \text{if}\ x = 0 \ , \\ \frac{\pi-x}{2} & \text{if} \ 0 < x < \pi \ . \end{cases} $$

How to Cite This Entry:
Discontinuous function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discontinuous_function&oldid=39786
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article