continuous

Let $X$ and $Y$ be topological spaces. A function $f\colon X\to Y$ is continuous if, for every open set $U\subset Y$ , the inverse image $f^{-1}(U)$ is an open subset of $X$ .

In the case where $X$ and $Y$ are metric spaces (e.g. Euclidean space, or the space of real numbers), a function $f\colon X\to Y$ is continuous if and only if for every $x\in X$ and every real number $\epsilon>0$ , there exists a real number $\delta>0$ such that whenever a point $z\in X$ has distance less than $\delta$ to $x$ , the point $f(z)\in Y$ has distance less than $\epsilon$ to $f(x)$ .

Continuity at a point

A related notion is that of local continuity, or continuity at a point (as opposed to the whole space $X$ at once). When $X$ and $Y$ are topological spaces, we say $f$ is continuous at a point $x\in X$ if, for every open subset $V\subset Y$ containing $f(x)$ , there is an open subset $U\subset X$ containing $x$ whose image $f(U)$ is contained in $V$ . Of course, the function $f\colon X\to Y$ is continuous in the first sense if and only if $f$ is continuous at every point $x\in X$ in the second sense (for students who haven’t seen this before, proving it is a worthwhile exercise).

In the common case where $X$ and $Y$ are metric spaces (e.g., Euclidean spaces), a function $f$ is continuous at $x\in X$ if and only if for every real number $\epsilon>0$ , there exists a real number $\delta>0$ satisfying the property that $d_{Y}(f(x),f(z))<\epsilon$ for all $z\in X$ with $d_{X}(x,z)<\delta$ . Alternatively, the function $f$ is continuous at $a\in X$ if and only if the limit of $f(x)$ as $x\to a$ satisfies the equation

\lim_{x\to a}f(x)=f(a).

Title	continuous
Canonical name	Continuous
Date of creation	2013-03-22 11:51:55
Last modified on	2013-03-22 11:51:55
Owner	djao (24)
Last modified by	djao (24)
Numerical id	12
Author	djao (24)
Entry type	Definition
Classification	msc 26A15
Classification	msc 54C05
Classification	msc 81-00
Classification	msc 82-00
Classification	msc 83-00
Classification	msc 46L05
Synonym	continuous function
Synonym	continuous map
Synonym	continuous mapping
Related topic	Limit
Defines	continuous at