Analysis is so hard...

I have a couple of questions here... Any help on how to start would be greatly appreciated!!

1- Let f: R->R be a strictly increasing function. Show that if the image of f is an interval then f is continuous.

2- Let S, T and U be metric spaces, T be compact, and g : T -> U be a continuous and one-to-one function. Prove that f : S -> T is uniformly continuous if and only if g ( f) : S -> U is uniformly continuous.