# Thread: Uniformly Continuous f and g functions

1. ## Uniformly Continuous f and g functions

I've been trying to figure out a couple of questions, with no luck. Any help would be really appreciated.

1. Let f and g be real valued functions that are uniformly continuous on D. Prove that f + g is uniformly continuous on D.

2. Let f and g be real valued functions that are uniformly continuous on D, and suppose that g(x) does not equal 0 for all x is an element of D.

a. Find an example to show that the function f/g need not be uniformly continuous on D.

b. Prove that if D is compact, then f/g must be uniformly continous on D.

Again, thank you to anyone for any help.

2. Originally Posted by QPerfection
I've been trying to figure out a couple of questions, with no luck. Any help would be really appreciated.

1. Let f and g be real valued functions that are uniformly continuous on D. Prove that f + g is uniformly continuous on D.
Just do it.

2. Let f and g be real valued functions that are uniformly continuous on D, and suppose that g(x) does not equal 0 for all x is an element of D.

a. Find an example to show that the function f/g need not be uniformly continuous on D.
Hint:

Spoiler:
Think about $\displaystyle D=(0,1),f(x)=x,g(x)=\cdots$

b. Prove that if D is compact, then f/g must be uniformly continous on D.
What's that one theorem about continuous functions on compact subspaces of $\displaystyle \mathbb{R}$?