<s>[INST] 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)
 [/INST] or every $x\in (-1,1).$ Define, 
$$ f(x) =100000+ 100x+ x^2\sin\frac{1}{x} \qquad\text{with $f(0)=100000$}$$
which is differentiable,  and 
$$ f'(x) =  100+ 2x\sin\frac{1}{x}-\cos\frac{1}{x}  \qquad\text{with $f'(0)=100$}$$
$f'$ is not continuous a $x=0$. Also $f'\ge 0$ indeed,
$$ f'(x)=100+2x\sin\frac{1}{x}-\cos\frac{1}{x} \ge 100+ 2x\sin\frac{1}{x}-1$$ 
Since $|\sin a| \le |a|$ and $-1\le-\cos a\le 1$ then $|2x\sin\frac{1}{x}|\le 2$ i.e $$2x\sin\frac{1}{x}\ge -2 $$
therefore 

$$ f'(x)=100+2x\sin\frac{1}{x}-\cos\frac{1}{x} \ge 100-2-1= 97>0$$ for every $x\in \mathbb R$. so $f $ is increasing and  $f'$ is not continuous at $x=0$.

Also $|x^2\sin\frac{1}{x}|\le 1$ for all $x\in (-1,1)$ so that 
$$f(x) =100000+ 100x+ x^2\sin\frac{1}{x} > 100000-100 -1 >0$$ for every $x\in (-1,1).$
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