Uniformly Continuous f and g functions

May 2010
1
0
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.
 

Drexel28

MHF Hall of Honor
Nov 2009
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Berkeley, California
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:

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 \(\displaystyle \mathbb{R}\)?