deepdream of
          a sidewalk
Show Answer
Math and science::Topology

Metric space. Function continuity

Function continuity between metric spaces

Let X and Y be metric spaces and let f:XY be a function. We say that f is continuous iff:

For all x0X and for all ε>0 there exists a [...] such that ([...] [...]).

Independence of metrics

The idea of continuity appears to depend on the notion of metric/distance as it is formulated using ε-balls. However, the following lemma reveals that this is not really so—all that is needed is the notion of open (or closed) subsets.

Equivance to function continuity

Let X and Y be metric spaces and let f:XY be a function. The the following three statements are equivalent:

  1. f is continuous;
  2. for all open UY, [...];
  3. for all closed VY, [...].

Informally: [2. can be rephrased as...].

Proof at the bottom of the back side.

This result motivates the definition of a topolgical space.