# Link between uniform continuity of two differentiable functions

I'm working on a question that states:

Let $$f,g:\mathbb{R}\rightarrow\mathbb{R}$$ be two differentiable functions with $$g'(x)\neq0$$ for all $$x\in\mathbb{R}$$. Consider the function $$h=\frac{f'}{g'}$$. Given that $$h$$ is bounded, what is the link between the following statements:

(a) $$f$$ is uniformly continuous

(b) $$g$$ is uniformly continuous

My first thought was that (b) implies (a), but not necessarily the other way around.

To prove that (b) implies (a), I noticed that since $$h$$ is bounded we have $$|f'(x)|\leq M|g'(x)|$$ for an $$M\in\mathbb{R}^+$$. Now using the fact that $$g$$ is uniformly continuous and the mean value theorem I showed that $$f$$ was also uniformly continuous. Is this correct?

The other way around I struggled to find a counterexample, but I was trying to do something with $$\sin$$ and $$\cos$$. Can anyone help with this?

Your proof for $$b\Rightarrow a$$ looks valid.
For a counterexample to the other direction, you could let $$g(x) = x^3+x$$ and $$f(x)=1$$.