# 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.

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## [edit] Formal definitions of continuity

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

### [edit] Function of a real variable

The 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 if (it is defined in a neighborhood of *x*_{0} and) for any there exist δ > 0 such that

Simply stated, the limit

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

### [edit] 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 if for all there exists δ > 0 such that

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

### [edit] 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 , is said to be **continuous** at the point if for every open set containing the point *y=f(x)*, there exists an open set containing *x* such that . Here . 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*).

## [edit] Continuous function

If the function *f* is continuous at every point 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 is said to be continuous if for any open set (respectively, closed subset of *Y* ) the set is an open set in *O*_{x} (respectively, a closed subset of *X*).

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