# Continuity

(Redirected from Continuous function)

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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

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

### 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  and) for any  there exist  such that



Simply stated, the limit



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

### Function on a metric space

A function f from a metric space  to another metric space  is continuous at a point  if for all  there exists  such that



If we let  denote the open ball of radius r round x in X, and similarly  denote the open ball of radius r round y in Y, we can express this condition in terms of the pull-back 



### Function on a topological space

A function f from a topological space  to another topological space , 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,  and  can be taken to be, respectively, a neighbourhood of x and a neighbourhood of .

## 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  (respectively, a closed subset of X).