Continuous function on metric spaces
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