Show that a homeomorphism preserves uniform continuity

[Ratpigeon]

Homework Statement

(X,d1),(Y,d2) and (Z,d3) are metric spaces, Y is compact,
g(y) is a continuous function that maps Y->Z with a continuous inverse
If f(x) is a function that maps X->Y, and h(x) maps X->Z such that h(x)=g(f(x))
Show that if h is uniformly continuous, f is uniformly continuous

Homework Equations

if h is continuous, f is continuous

The Attempt at a Solution

I proved that f is continuous when h is continuous.
I also know that g is a homeomorphism, which preserves pretty much everything.
and that f(x)=g^(-1)(h(x)) so I just need to show that the homeomorphism preserves the uniform continuity.

 

[micromass]

What do you know about continuous functions with compact domain?

 

[Ratpigeon]

THey have a compact range. But I don't know that f's domain is compact, only that it's range is...

 

[micromass]

g has a compact domain.

 

[Ratpigeon]

I know Y is compact, and so g(Y) is compact.
h(X) c g(Y), so h is bounded. Also, because the range of f is compact, f is bounded.
I qualitatively know that uniform continuity means that the function is not allowed to get infinitely steep, and that a continuous, bounded function obviously can't be infinitely steep, therefore it should be uniformly continuous, but I'm not sure how to say it mathematically.

 

[micromass]

Can you show that a continuous function on a compact domain is uniform continuous?

 

[Ratpigeon]

Ooooh...
Thanks. I've got it now.