# Uniformly Continuous f and g functions

• May 12th 2010, 10:47 PM
QPerfection
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.
• May 12th 2010, 11:02 PM
Drexel28
Quote:

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.

Quote:

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 $D=(0,1),f(x)=x,g(x)=\cdots$

Quote:

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 $\mathbb{R}$?