Does a bounded function converge if its derivative tends to zero?

real-analysis derivatives

First idea: If you drop the boundedness assumption, $\ln(x)$ does provide a counterexample to what you want.

Now think about how you can transform this counterexample into a bounded counterexample.

Hint

Typical bounded functions include $\sin$ and $\cos$.


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.

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