Boundedness of derivative of bounded, monotonic, continuously differentiable function

Question

Let $f\in C^1(\mathbb{R})$ be bounded and monotonic. What else do we need from $f$ for its derivative $f'$ to be bounded, too?

Answer 1

A differentiable function has bounded derivative if and only if it is Lipschitz-continuous. I don't think one can say more than that because you can always have arbitrarily steep spikes on arbitrarily short intervals so that $f$ remains bounded and monotone.

Answer 2

The "Single Spike" example:

Consider $f_n(x)=x^n$, $x\in[0,1], n\in N$. Each $f_n$ is monotonous and bounded, but $\{f_n'(1)=n\}$ not limited.