Is a monotone differentiable function continuously differentiable? If $f:\mathbb{R}^+\to\mathbb{R}$ is monotonic and differentiable, does it follow that $f$ is continuously differentiable? (This question arose from discussion here: problem on continuous and differentiable function)

Let $f(x)=x|x|(\sin(1/x))+2x|x|+2x$ for $x\
e 0.$ Let $f(0)=0.$ Then:

* For $x > 0$, $f'(x)=2x \sin(1/x)-\cos(1/x)+4x+2=2x(\sin (1/x)+2)+2-\cos(1/x)$, which is positive for $x>0$.
* For $x < 0$, $f'(x)=-2x \sin(1/x)+\cos(1/x)-4x+2 =-2x(\sin (1/x) + 2) + 2 + \cos(1/x)$, which is positive for $x<0$.
* We can evaluate $f'(0)$ directly: $$f'(0)=\lim_{x \to 0} \frac{f(x)}{x}=\lim_{x \to 0} \left(|x|\sin (1/x) + 2|x| + 2\right)=2 \, . $$



So $f$ is everywhere differentiable with positive derivative; thus it is monotone. But $f'$ is discontinuous at $0$.