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I was trying to help this person with his question.

Integrable but not differentiable function

Showing continuity was relatively easy, but I'm having trouble with showing that the function is not differentiable at $c$.

I tried using the hint and using one-sided derivatives, but all I got was that $F'(x) = f(x)$ for all $x\neq c$, and all I managed to show from that was that $\lim_\limits{x \to c} F'(x)$ does not exist. But all that showed was that the derivative of $F$ wasn't continuous at $c$, not that $F$ wasn't differentiable at $c$.

So I tried going down to Riemann sums. But I got stuck when showing that the last bit was a contradiction, since the absolute value of an integral is at most the integral of the absolute value of the integrand, not the other way round. I was wondering if it's salvageable, or if there are other errors I didn't see. Thanks!

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Ilham
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    Try showing that one-sided derivatives of F exist, but are not equal – Wojowu Apr 22 '15 at 20:55
  • @I'm having trouble with that, I had an edit before that showed that if $f$ is right-continuous at $c$, then $F$ is right differentiable at $c$ and the the derivative from the right is $\lim_{x\to c^+} f(x)$. But then I ran into trouble with the details when showing that $F$ must have unequal left and right derivatives. – Ilham Apr 22 '15 at 20:59
  • What trouble? The left derivative has to be $\lim_{x\to c^-}f(x).$ By definition of a jump discontinuity, this is not equal to the limit from the right. – zhw. Apr 23 '15 at 00:51
  • @zhw. How do I prove that? I get that the left derivative of $F$ at $c$ is:

    $$\lim_\limits{x\to c^-} \frac{F(x)-F(c)}{x-c} =\lim_\limits{x\to c^-} \frac{\int_c^x f(t) dt}{x-c} $$ but how is the that equal to $$\lim_\limits{x\to c^-} f(x)$$?

     – Ilham Apr 23 '15 at 19:00
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    Define $g = f$ on $[a,c), g(c) = \lim_{x\to c^-}f(x).$ Then $g$ is continuous on $[a,c].$ Note $F(x)-F(c) = \int_c^x g.$ Because $g$ is continuous, you know the left derivative of $F$ at $c$ is $g(c).$ – zhw. Apr 23 '15 at 19:13
  • Thank you! I can finally finish that question. – Ilham Apr 23 '15 at 19:31

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