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[edit]

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

Function of a real variable[edit]

The ${\displaystyle \delta -ϵ}$ 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 ${\displaystyle x_0\in \mathbb{ℝ}}$ if (it is defined in a neighborhood of ${\displaystyle x_0}$ and) for any ${\displaystyle \epsilon >0}$ there exist ${\displaystyle \delta >0}$ such that

${\displaystyle |x-x_0|<\delta ⟹|f(x)-f(x_0)|<\epsilon .}$

Simply stated, the limit

${\displaystyle \underset{x\to x_0}{\lim }f(x)=f(x_0).}$

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

Function on a metric space[edit]

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

${\displaystyle d(x,x_0)<\delta ⟹e(f(x),f(x_0))<\epsilon .}$

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

${\displaystyle f^{⊣}[B_e(f(x),\epsilon )]\supseteq B_d(x,\delta ).}$

Function on a topological space[edit]

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

Continuous function[edit]

If the function f is continuous at every point ${\displaystyle x\in X}$ 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 ${\displaystyle f:(X,O_X)\to (Y,O_Y)}$ is said to be continuous if for any open set ${\displaystyle U\in O_Y}$ (respectively, closed subset of Y ) the set ${\displaystyle f^{-1}(U)=\{x\in X\mid f(x)\in U\}}$ is an open set in ${\displaystyle O_x}$ (respectively, a closed subset of X).