Is it true or false that for any two uniformly continuous functions f and g on a set D such that g(x)=/=0 on all of D the function f/g is uniformly continuous on D? If yes, prove, if not show a counterexample. What if D is compact?

Thanks!

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- Nov 4th 2010, 10:36 PMAKTiltedf/g if both are uniformly continuous
Is it true or false that for any two uniformly continuous functions f and g on a set D such that g(x)=/=0 on all of D the function f/g is uniformly continuous on D? If yes, prove, if not show a counterexample. What if D is compact?

Thanks! - Nov 4th 2010, 11:04 PMroninpro
How about $\displaystyle f(x)=1$ and $\displaystyle g(x)=x$, defined on $\displaystyle (0,\infty)$? Those two functions are uniformly continuous, but the function $\displaystyle f(x)/g(x)=1/x$ is not.

For the second question, you should ask yourself if there are any properties for continuous functions on compact sets.