# Show that a homeomorphism preserves uniform continuity

1. May 2, 2012

### Ratpigeon

1. The problem statement, all variables and given/known data

(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

2. Relevant equations

if h is continuous, f is continuous

3. 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.

2. May 2, 2012

### micromass

Staff Emeritus
What do you know about continuous functions with compact domain?

3. May 2, 2012

### Ratpigeon

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

4. May 2, 2012

### micromass

Staff Emeritus
g has a compact domain.

5. May 3, 2012

### 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.

6. May 3, 2012

### micromass

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

7. May 3, 2012

### Ratpigeon

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