Let $f$ be Riemann integrable on $[a, b]$, let $c\in(a, b)$, and let
$\displaystyle F(x)=\int_a^x f(t)\ dt$, $a\le x\le b$. For the following statement, give either a proof or a counterexample:
If $f$ is differentiable at $c$, then $F'$ is continuous at $c$.
My attempt:
My hunch is that the statement is false. Since $f$ is differentiable at $c$, $f$ is continuous at $c$ and so by the first Fundamental Theorem of Calculus, $F'(c)=f(c)$. But $F'$ need not be equal to $f$ at other points.
Is this correct? If so, how do I find a counterexample? If not, how do I prove the statement?
Best Answer
Construct a counterexample where $F'$ fails to exist at infinitely many points $x_n \to c$ and is technically not continuous at $c$, but where $f'(c)$ exists.
For example, take $c= 0$ and $f:[0,1/2] \to \mathbb{R}$ such that $f(0) = 0$ and for $x \in (0,1/2]$ and $k = 1,2, \ldots$
$$f(x) = x^2 \begin{cases}\,\,\,\,\,1, \,\,\,\, x\in \left(\frac{1}{2k+1},\frac{1}{2k}\right] \\ -1, \,\,\,\, x\in \left(\frac{1}{2k+2},\frac{1}{2k+1}\right] \end{cases}$$