# Uniform Continuity Proof

• Nov 1st 2010, 06:42 PM
AKTilted
Uniform Continuity Proof
Hi,
I'm not sure how to prove this statement:

Suppose f is uniformly continuous on the set S and let T be the image of S under f. Also assume that the function g is uniformly continuous on T. Prove that g o f is uniformly continuous on S?

Thanks a lot!
• Nov 1st 2010, 06:46 PM
Drexel28
Quote:

Originally Posted by AKTilted
Hi,
I'm not sure how to prove this statement:

Suppose f is uniformly continuous on the set S and let T be the image of S under f. Also assume that the function g is uniformly continuous on T. Prove that g o f is uniformly continuous on S?

Thanks a lot!

So, you're trying to show that given $\displaystyle \varepsilon>0$ there exists $\displaystyle \delta>0$ such that for all $\displaystyle x,y\in S$ $\displaystyle |x-y|<\delta\implies |f(g(x))-f(g(y))|<\varepsilon$, right?

Ok, so what's the first step? You're lcearly going to want to consider two deltas one for the $\displaystyle S$ and one for $\displaystyle T$. Which should you pick first?