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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?

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Your proof for $b\Rightarrow a$ looks valid.

For a counterexample to the other direction, you could let $g(x) = x^3+x$ and $f(x)=1$.

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