Continuity

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Formal definitions of continuity

We can develop the definition of continuity from the $\delta-\epsilon$ formalism which are usually taught in first year calculus courses to general topological spaces.

Function of a real variable

The $\delta-\epsilon$ formalism defines limits and continuity for functions which map the set of real numbers to itself. To compare, we recall that at this level a function is said to be continuous at $x_0\in\mathbb{R}$ if (it is defined in a neighborhood of $x_0$ and) for any $\varepsilon>0$ there exist $\delta>0$ such that

$|x-x_0| < \delta \implies |f(x)-f(x_0)| < \varepsilon. \,$

Simply stated, the limit

$\lim_{x\to x_0} f(x) = f(x_0).$

This definition of continuity extends directly to functions of a complex variable.

Function on a metric space

A function f from a metric space $(X,d)$ to another metric space $(Y,e)$ is continuous at a point $x_0 \in X$ if for all $\varepsilon > 0$ there exists $\delta > 0$ such that

$d(x,x_0) < \delta \implies e(f(x),f(x_0)) < \varepsilon . \,$

If we let $B_d(x,r)$ denote the open ball of radius r round x in X, and similarly $B_e(y,r)$ denote the open ball of radius r round y in Y, we can express this condition in terms of the pull-back $f^{\dashv}$

$f^{\dashv}[B_e(f(x),\varepsilon)] \supseteq B_d(x,\delta) . \,$

Function on a topological space

A function f from a topological space $(X,O_X)$ to another topological space $(Y,O_Y)$ , usually written as $f:(X,O_X) \rightarrow (Y,O_Y)$ , is said to be continuous at the point $x \in X$ if for every open set $U_y \in O_Y$ containing the point y=f(x), there exists an open set $U_x \in O_X$ containing x such that $f(U_x) \subset U_y$ . Here $f(U_x)=\{f(x') \in Y \mid x' \in U_x\}$ . In a variation of this definition, instead of being open sets, $U_x$ and $U_y$ can be taken to be, respectively, a neighbourhood of x and a neighbourhood of $y=f(x)$ .

Continuous function

If the function f is continuous at every point $x \in X$ then it is said to be a continuous function. There is another important equivalent definition that does not deal with individual points but uses a 'global' approach. It may be convenient for topological considerations, but perhaps less so in classical analysis. A function $f:(X,O_X) \rightarrow (Y,O_Y)$ is said to be continuous if for any open set $U \in O_Y$ (respectively, closed subset of Y ) the set $f^{-1}(U)=\{ x \in X \mid f(x) \in U\}$ is an open set in $O_x$ (respectively, a closed subset of X).