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

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.