Uniform Continuity Proof

**AKTilted** · November 1st 2010, 06:42 PM

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!

**Drexel28** · November 1st 2010, 06:46 PM

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 $\varepsilon>0$ there exists $\delta>0$ such that for all $x,y\in S$ $|x-y|<\delta\implies |f(g(x))-f(g(y))|<\varepsilon$ , right?

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

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