Continuity

In mathematics, the notion of continuity of a function relates to the idea that the "value" of the function should not jump abruptly for any vanishingly "small" variation to its argument. Another way to think about a continuity of a function is that any "small" change in the argument of the function can only effect a correspondingly "small" change in the value of the function.

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).