TITLE: Does a bounded function converge if its derivative tends to zero?
QUESTION [5 upvotes]: Suppose we have function $f:\Bbb R^+\to\Bbb R$ that is bounded $|f(t)|<M$ and differentiable such that $\lim\limits_{t\to\infty}f'(t) = 0$. Does this imply that $f(t)$ converges?

REPLY [2 votes]: In general the answer is NOT. Let's consider the function $f(t) = \sin(\ln(t))$. Its derivative $f' = \cos(\ln(t))/t$ converges to $0$ but in limit, the function keeps oscillating.