4

Given a function $f$ that is differentiable at a point $c$, if we define (using the Riemann integral)

$$F(x) = \int_a^x f(t)dt.$$

I want to determine If $f$ is differentiable at $c$ then $F'$ is continuous at $c$.

This is apparently false, since I cannot guarantee that $F'$ exists in a neighborhood around $c$. However, I can't think of any counterexample to argue my idea, any suggestions?

Thanks

Gary
  • 31,845
Wrloord
  • 1,626
  • 1
    @EeveeTrainer It is not givren that $f$ is continuous. If it is just RI you cannot say $F'=f$. – Kavi Rama Murthy Dec 14 '21 at 23:11
  • $F'$ need not even exist in any neighborhood of $c$, so the result is false. – Kavi Rama Murthy Dec 14 '21 at 23:14
  • @EeveeTrainer : Differentiability of $f$ (not of $F$) implies continuity of $f,$ and continuity of $f$ implies differentiability of $F,$ but the question is about continuity of $F',$ not about mere existence of $F'. \qquad$ – Michael Hardy Dec 14 '21 at 23:26
  • f being differentiable at c alone does not mean its integral from a to x exists. Please supply more context. – Suzane Dec 15 '21 at 00:56
  • 3
    Try something like this (check whether the details work): Define $f:[0,1]\to\mathbb R$ by $f(0)=0$, on $(\frac{1}{n+1}, \frac{1}{n}]$ take the value $\frac{(-1)^n}{n^2}$ for each $n=1,2,3,\dots$. Then $f'(0)=0$ but $F'(\frac1n)$ cannot exist (because of the jump discontinuities. Did it work? – B. S. Thomson Dec 15 '21 at 02:09
  • @KaviRamaMurthy "$F'$ need not even exist in any neightborhood of $c$." Careful here. If $f$ is Riemann integrable then $F'$ exists almost everywhere. What you mean (and the OP means) is that $F'$ need not exist at every point of some neighborhood of $c$. – B. S. Thomson Dec 15 '21 at 02:28
  • @B.S.Thomson Interesting counterexample, that means that if $f(x)$ has jump discontinuities it is not guaranteed that $F′=f$ ? – Wrloord Dec 15 '21 at 03:21
  • 3
    Yes; a derivative cannot have jump discontinuities. This is. Darboux's theorem, see e.g. https://math.stackexchange.com/questions/4326051/implications-of-darbouxs-theorem-when-f-is-differentiable-in-some-interval-c – Calvin Khor Dec 15 '21 at 03:29
  • See https://math.stackexchange.com/a/3516923/72031 – Paramanand Singh Feb 17 '22 at 07:42

0 Answers0