Topology MOC

Continuity

In its most general form, a function between topological spaces is continuous^[German stetig in ] at a point iff. for every (open) neighbourhood of , there exists an (open) neighbourhood of , such that .¹ topology Intuitively, you can move a small amount in any direction from and end up close to .

A function is continuous iff. it is continuous at every point in its domain, or equivalently iff. the preïmage of every open set is open. topology Category of topological spaces has such functions as its morphisms. We write to refer to the^[well, a] Function space of continuous functions on .

Proof of equivalence of open and general neighbourhood pointwise definitions

Let and be topological spaces, and .

First assume for every open neighbourhood of , there exists an open neighbourhood of , such that . Given an arbitrary neighbourhood of , there exists an open neighbourhood such that . Thence there exists an open neighbourhood of , such that . Therefore for every neighbourhood of , there exists a neighbourhood of , such that .

For the converse, assume for every neighbourhood of , there exists a neighbourhood of , such that . Let be an open neighbourhood of . Then there exists a neighbourhood of such that . It follows there exists an open neighbourhood of , such that . Therefore for every open neighbourhood of , there exists an open neighbourhood of , such that .

Proof of equivalence of pointwise and preïmage definition

Let and be topological spaces, and .

First assume the preïmage of every open set is open. Let some , and be an open neighbourhood of . The preïmage is then an open neighbourhood of , and (image of preïmage). Therefore, given any and any open neighbourhood of , there exists an open neighbourhood of such that .

For the converse, assume given any and any open neighbourhood of , there exists an open neighbourhood of such that . Let be an open set. For every , let be an open neighbourhood of such that . Take the union , which is an open neighbourhood of every , whence Since every it follows that , whence . Thus is open. Therefore the preïmage of every open set is open.

A continuous bijection with a continuous inverse is a Homeomorphism².

Special cases

In a metric space

If and are metric spaces then the definition may be restated as

A function is continuous at a point iff. for every there exists such that , i.e. for any .

In metric spaces continuity is equivalent to Sequential continuity, namely a function is continuous at a point iff. it is sequentially continuous at that point.

In the real numbers

Intuitively, a function is continuous if it has no gaps, i.e. for you can sketch the function without the pen leaving the page. More formally continuity is defined in terms of Limits (Calculus). A function is continuous at iff.

and is itself continuous iff. it is continuous at all points in its domain.

A function which is differentiable at is continuous at , but the converse is not necessarily true

Hypernyms include

• Piecewise continuous
• Left and right continuity

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Footnotes

1. Using the notation of an Image. Can be restates as for any . ↩

2. Not to be confused with homomorphism. ↩