# Continuous function on metric spaces

Submitted by Structure on Mon, 11/19/2007 - 20:28.

Let be a metric space where X is a set and is the distance, i.e.

d has properties

and if and only if .

There is a canonical topology on namely such as a separated topological space. To describe the topology of it is useful to introduce a ball of center and radius r, i.e. the set

Now a set is open if and only if for all there is a such as .

Now we can add an equivalent condition to theorem "Continuous Function" about continuous function in this special case of topological metric space.

Let and metric spaces and

Then f is continuous on X if and only if

such as if we have