Continuity

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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 ${\displaystyle \delta -\epsilon }$ formalism which are usually taught in first year calculus courses to general topological spaces.

Function of a real variable[edit]

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

${\displaystyle |x-x_{0}|<\delta \implies |f(x)-f(x_{0})|<\varepsilon .\,}$

Simply stated, the limit

${\displaystyle \lim _{x\to x_{0}}f(x)=f(x_{0}).}$

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

Function on a metric space[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 ${\displaystyle \varepsilon >0}$ there exists ${\displaystyle \delta >0}$ such that

${\displaystyle d(x,x_{0})<\delta \implies e(f(x),f(x_{0}))<\varepsilon .\,}$

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 ${\displaystyle f^{\dashv }}$

${\displaystyle f^{\dashv }[B_{e}(f(x),\varepsilon )]\supseteq B_{d}(x,\delta ).\,}$

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 ${\displaystyle f:(X,O_{X})\rightarrow (Y,O_{Y})}$, 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 ${\displaystyle f(U_{x})=\{f(x')\in Y\mid x'\in U_{x}\}}$. 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 ${\displaystyle f:(X,O_{X})\rightarrow (Y,O_{Y})}$ 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).