Continuity

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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

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.

## Contents

* 1 Formal definitions of continuity
* 1.1 Function of a real variable
* 1.2 Function on a metric space
* 1.3 Function on a topological space
* 2 Continuous function

## Formal definitions of continuity[edit]

We can develop the definition of continuity from the δ−ϵ formalism which are usually taught in first year calculus courses to general topological spaces.

### Function of a real variable[edit]

The δ−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 x0∈ℝ if (it is defined in a neighborhood of x0 and) for any ε>0 there exist δ>0 such that

|x−x0|<δ⟹|f(x)−f(x0)|<ε.

Simply stated, the limit

limx->x0f(x)=f(x0).

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 (X,d) to another metric space (Y,e) is continuous at a point x0∈X if for all ε>0 there exists δ>0 such that

d(x,x0)<δ⟹e(f(x),f(x0))<ε.

If we let Bd(x,r) denote the open ball of radius r round x in X, and similarly Be(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⊣

f⊣[Be(f(x),ε)]⊇Bd(x,δ).

### Function on a topological space[edit]

A function f from a topological space (X,OX) to another topological space (Y,OY), usually written as f:(X,OX)->(Y,OY), is said to be continuous at the point x∈X if for every open set Uy∈OY containing the point y=f(x), there exists an open set Ux∈OX containing x such that f(Ux)⊂Uy. Here f(Ux)={f(x′)∈Y∣x′∈Ux}. In a variation of this definition, instead of being open sets, Ux and Uy can be taken to be, respectively, a neighbourhood of x and a neighbourhood of y=f(x).

## Continuous function[edit]

If the function f is continuous at every point x∈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,OX)->(Y,OY) is said to be continuous if for any open set U∈OY (respectively, closed subset of Y ) the set f−1(U)={x∈X∣f(x)∈U} is an open set in Ox (respectively, a closed subset of X).