Prove or Disprove: Let $f:\mathbb{R} \to \mathbb{R}$ be a bounded uniformly continuous function that whose first and second derivative exists and is continuous, in other words $f \in C^2_{unif} (\mathbb{R},\mathbb{R})$. Then $f'(x)$ is bounded.
This is a problem that I came across while working on a project. At first we felt that it wasn't true but we've been unable to find a counterexample. Any advice on how to prove it (if it is true) would be much appreciated. I apologize if this has been posted before.
It's not true, as a counter example take a sine curve with decreasing amplitude but frequency increasing to $\infty$ (this will mean unbounded derivative). Something like:
$$\frac{1}{1+x^2}\sin(x^5)$$