<s>[INST] Is there a differentiable function f which the differential function f' is bounded but has no maximum on a closed interval. Is there a differentiable function $f$ in which the differential function $f'$ is bounded but has no maximum on one closed interval? Thanks 
 [/INST] I found this example in "An introduction to Analysis"(in Japanese) by Ichiro Tajima.
$$f(x)=\begin{cases} x^2 \sin (\frac{1}{x} + 3 x), \  x\in [-\frac{1}{3}, 0) \cup (0, \frac{1}{3}]\\ 0, \ x = 0\end{cases}.$$
$$f'(x)=\begin{cases} 2 x \sin (\frac{1}{x} + 3 x) + (3 x^2 - 1) \cos (\frac{1}{x} + 3 x), \  x\in [-\frac{1}{3}, 0) \cup (0, \frac{1}{3}]\\ 0, \ x = 0\end{cases}.$$
If $x \neq 0$, then $$f'(x) = \sqrt {1 - 2 x^2 + 9 x^4} \sin (\frac{1}{x} + 3 x + \alpha(x)).$$
And $g(x) := 1 - 2 x^2 + 9 x^4$ attains the maximum value at $x = 0$.
So, $$\{f'(x) \mid x \in [-\frac{1}{3}, \frac{1}{3}]\} = (-1, 1).$$

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