View source for Continuous function

A basic concept in mathematical analysis.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c0256501.png" /> be a real-valued function defined on a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c0256502.png" /> of the real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c0256503.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c0256504.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c0256505.png" /> is said to be continuous at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c0256506.png" /> (or, in more detail, continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c0256507.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c0256508.png" />) if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c0256509.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565010.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565011.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565012.png" /> the inequality

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is valid. If one denotes by

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and

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the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565016.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565017.png" />-neighbourhoods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565019.png" />, respectively, then the definition above can be rephrased as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565020.png" /> is called continuous at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565021.png" /> if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565022.png" />-neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565024.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565025.png" />-neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565027.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565028.png" />.

By using the concept of a limit one can say that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565029.png" /> is continuous at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565030.png" /> if its limit with respect to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565031.png" /> exists at that point and if this limit is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565032.png" />:

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This is equivalent to

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565037.png" />; that is, to an infinitely small increment of the argument at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565038.png" /> corresponds an infinitely small increment of the function.

In terms of the limit of a sequence, the definition of continuity of a function at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565039.png" /> is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565040.png" /> is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565041.png" /> if for every sequence of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565043.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565044.png" />, one has

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All these definitions of a function being continuous at a point are equivalent.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565046.png" /> is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565047.png" /> with respect to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565048.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565049.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565050.png" /> is said to be continuous on the right (or left) at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565051.png" />.

All basic [[Elementary functions|elementary functions]] are continuous at all points of their domains of definition. An important property of continuous functions is that their class is closed under the arithmetic operations and under composition of functions. More accurately, if two real-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565054.png" />, are continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565055.png" />, then so is their sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565056.png" />, difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565057.png" /> and product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565058.png" />, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565059.png" />, also their quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565060.png" /> (which is necessarily defined in the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565061.png" /> with a certain neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565062.png" />). If, as before, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565063.png" /> is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565066.png" />, is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565067.png" />, so that the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565068.png" /> makes sense, if there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565069.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565070.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565071.png" /> is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565072.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565073.png" /> is also continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565074.png" />. Thus, in this case

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that is, in this sense the operation of limit transition commutes with the operation of applying a continuous function. From these properties of continuous functions it follows that not only the basic, but also arbitrary elementary functions are continuous in their domains of definition. The property of continuity is also preserved under a uniform limit transition: If a sequence of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565076.png" /> converges uniformly on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565077.png" /> and if every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565078.png" /> is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565080.png" /> then

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is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565082.png" />.

If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565083.png" /> is continuous at every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565084.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565085.png" /> is said to be continuous on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565086.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565088.png" /> is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565089.png" />, then the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565090.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565091.png" /> is also continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565092.png" />. The converse is not true, in general. For example, the restriction of the [[Dirichlet-function|Dirichlet function]] either to the set of rational numbers or to the set of irrational numbers is continuous, but the Dirichlet function itself is discontinuous at all points.

An important class of real-valued continuous functions of a single variable consists of those functions that are continuous on intervals. They have the following properties.

Weierstrass' first theorem: A function that is continuous on a closed interval is bounded on that interval.

Weierstrass' second theorem: A function that is continuous on a closed interval assumes on that interval a largest and a smallest value.

Cauchy's intermediate value theorem: A function that is continuous on a closed interval assumes on it any value between those at the end points.

The inverse function theorem: If a function is continuous and strictly monotone on an interval, then it has a single-valued inverse function, which is also defined on an interval and is strictly monotone and continuous on it.

Cantor's theorem on uniform continuity: A function that is continuous on a closed interval is uniformly continuous on it.

Every function that is continuous on a closed interval can be uniformly approximated on it with arbitrary accuracy by an algebraic polynomial, and every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565093.png" /> that is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565094.png" /> and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565095.png" /> can be uniformly approximated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565096.png" /> with arbitrary accuracy by trigonometric polynomials (see [[Weierstrass theorem|Weierstrass theorem]] on the approximation of functions).

The concept of a continuous function can be generalized to wider forms of functions, above all, to functions of several variables. The definition above is preserved formally if one understands by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565097.png" /> a subset of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565098.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c02565099.png" />, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650100.png" /> the distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650102.png" />, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650103.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650104.png" />-neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650105.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650106.png" />, and by

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the limit of a sequence of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650108.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650110.png" />, of several variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650111.png" /> that is continuous at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650112.png" /> is also called continuous at this point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650113.png" /> jointly in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650114.png" />, in contrast to functions of several variables that are continuous in the variables individually. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650115.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650116.png" />, is called continuous at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650117.png" /> in, say, the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650118.png" /> if the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650119.png" /> to the set

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is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650121.png" />, that is, if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650122.png" /> of the single variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650123.png" /> is continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650124.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650127.png" />, can be continuous at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650128.png" /> in every variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650129.png" />, but need not be continuous at this point jointly in the variables.

The definition of a continuous function goes over directly to complex-valued functions. Only one has to interpret <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650130.png" /> in the definition above as the norm of the complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650131.png" /> and

<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/c/c025/c025650/c025650132.png" /></td> </tr></table>

as the limit in the complex plane.

All these definitions are special cases of the more general concept of a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650133.png" /> with as domain of definition a certain topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650134.png" /> and with values in a certain topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650135.png" /> (see [[Continuous mapping|Continuous mapping]]).

Many properties of real-valued continuous functions of a single variable carry over to continuous mappings between topological spaces. A generalization of Weierstrass' theorem mentioned above: The continuous image of a compact topological space in a Hausdorff space is compact. A generalization of Cauchy's intermediate value theorem for a continuous function on a closed interval: A continuous image of a connected topological space in a topological space is also connected. A generalization of the theorem on the inverse function of a strictly monotone continuous function: A continuous one-to-one mapping of a compactum onto a Hausdorff space is a homeomorphism. A generalization of the theorem on the limit of a uniformly-convergent sequence of continuous functions: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650136.png" /> is a uniformly-convergent sequence of mappings of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650137.png" /> into a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650138.png" /> that are continuous (at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650139.png" />) then the limit mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650140.png" /> is also continuous (at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025650/c025650141.png" />). A generalization of Weierstrass' theorem on the approximation of functions that are continuous on a closed interval is the [[Stone–Weierstrass theorem|Stone–Weierstrass theorem]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,   "A course of mathematical analysis" , '''1–2''' , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" , '''1–2''' , MIR  (1982)  (Translated from Russian)</TD></TR></table>



====Comments====
The topics discussed above can be found in almost any introductory book on real analysis. Proofs of all statements can be found in, e.g., [[#References|[a5]]], Chapt. 3.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,   "Mathematical analysis" , Addison-Wesley  (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.G. Bartle,   "The elements of real analysis" , Wiley  (1976)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.H. Hardy,   "A course of pure mathematics" , Cambridge Univ. Press  (1975)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 75–78</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K.R. Stromberg,   "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R.P Boas jr.,   "A primer of real functions" , Math. Assoc. Amer.  (1960)</TD></TR></table>