
Uniform Continuity Again
If f(x) = x and g(x) = sin(x), show that f and g are uniformly continuous on the Reals, but their product, fg, is not uniformly continuous.
I can prove that f(x) is uniformly continuous, but I'm having difficulty proving that g(x) is uniformly continuous. Any help is much appreciated.

Same hint as on the previous thread (more or less). If a function has an unbounded derivative then it's not going to be uniformly continuous, because it's changing too rapidly. So differentiate the function to look for places where the derivative of this function is large.
For example if x is a large multiple of π and y is very close to x then f(y) can be very different from f(x).

Unfortunately, I'm not allowed to use derivatives.