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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?

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  • $\begingroup$ 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. $\endgroup$ – amsmath Sep 7 '19 at 15:00
  • $\begingroup$ @amsmath Edited. Please recheck it. $\endgroup$ – Topo Sep 7 '19 at 15:03
  • $\begingroup$ Check out my example. $f(x)$ is zero for negative $x$ and otherwise a parabola. $\endgroup$ – amsmath Sep 7 '19 at 15:04
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Define $f:[-1,1]\to\mathbb R$ by $$ f(x) = \begin{cases} 0&\text{for $x<0$}\\ x^2&\text{for $x\ge 0$} \end{cases}. $$

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