# uniform continuity problems

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• Nov 2nd 2009, 06:33 PM
dannyboycurtis
uniform continuity problems
I need help solving the following:
Assume f and g are uniformly continuous on D.

First I need to prove that if f and g are both bounded on D, that fg is uniformly continuous on D.

I also need to prove that if D is closed and bounded, then fg is uniformly continuous.

Also, if $g(x)\neq 0$ for any x in D and D closed and bounded, then $\frac{f}{g}$ is uniformly continuous.
Any help on these is greatly appreciated thanks!
• Nov 2nd 2009, 08:04 PM
redsoxfan325
Quote:

Originally Posted by dannyboycurtis
I need help solving the following:
Assume f and g are uniformly continuous on D.

First I need to prove that if f and g are both bounded on D, that fg is uniformly continuous on D.

I also need to prove that if D is closed and bounded, then fg is uniformly continuous.

Also, if $g(x)\neq 0$ for any x in D and D closed and bounded, then $\frac{f}{g}$ is uniformly continuous.
Any help on these is greatly appreciated thanks!

This inequality should help you:

$|f(x)g(x)-f(y)g(y)|\leq|f(x)||g(x)-g(y)|+|g(y)||f(x)-f(y)|$

For the second one, use the fact that D is closed and bounded and that f and g are u.c. to deduce that they are bounded and then it's reduced to part 1.