# Does there exists a one time differentiable function?

Does there exists a only one time differentiable function $$f$$ in an interval $$[a,b]$$ whose derivative is monotone increasing in $$[a,b]$$ ? That is, does there exists a function $$f$$ in $$[a,b]$$ such that $$f'$$ exists for all $$x\in [a,b]$$ and $$f'$$ is monotone increasing in $$[a,b]$$ but $$f''$$ does not exists at some point in $$[a,b]$$.

I guess there exists such a function. I'm trying to define such function but unable!!

Any hint?

• If a function is non-decreasing, its derivative exists almost everywhere... But take the function $f'(x) = \chi_{[0,\infty)}x$ (i.e., $f(x) = \int_0^x\chi_{[0,\infty)}(t)t\,dt$). That's an example. – amsmath Sep 7 '19 at 15:00
• @amsmath Edited. Please recheck it. – Topo Sep 7 '19 at 15:03
• Check out my example. $f(x)$ is zero for negative $x$ and otherwise a parabola. – amsmath Sep 7 '19 at 15:04

Define $$f:[-1,1]\to\mathbb R$$ by $$f(x) = \begin{cases} 0&\text{for x<0}\\ x^2&\text{for x\ge 0} \end{cases}.$$