Homeomorphism and identity

Let and Prove the following:

and are homeomorphisms, where is the euclidian distance, and distance of the maximum.

is supposed to be the identity function, but I'm confused how to work with.

is continuous iff is continuous for

The left implication is easy because if each is continuous then clearly is continuous, but don't know how to prove the right implication.

Re: Homeomorphism and identity

Topologically, the meaning of "the identity map on X is a homeomorphism" is that two seemingly different topologies are actually the same topology.

In your case, you'll have different metrics, but you'll show that the identity is a homeomorphism, meaning that despite the different ways of measuring distances, the topologies those metrics induce (via their open balls) are the same. The way you show that is to show that every open ball in one metric is open in the topology induced by the other metric (but perhaps not an open BALL anymore), and visa versa.