fellow mathematicians. I am trying to establish the following result:
Let $f:[a,b]\to\mathbb{R}$ be integrable in $[a,b]$ and let $c\in(a,b)$. Consider the function $$ F(x):=\int_{a}^{x}{f(t)\,dt}\,\text{such that}\,x\in[a,b] $$
If $f$ is differentiable at $c$, then $F'$ is continuous at $c$.
So far I've worked this out:
Since $f'(c)$ exists, $f$ is differentiable at $c$, then $f$ is continuous at $c$. By the Fundamental Theorem of Calculus $F(x)$ is continuous and $F$ is differentiable at $c$, moreover:
$$ F'(c)=f(c) $$
Now, $F(x)$ is a continuous function and is differentiable at a point $c$. Does that mean that $F(x)$ is differentiable in a neighbourhood of $c$? If that is the case, I could use the epsilon-delta definition of continuity to prove that $F'(x)$ is continuous at $c$. For example:
$F'(x)$ is continous at $c$ if the following is satisfied: $$ \lim_{x\to c}{F'(x)}=F'(c) \Longleftrightarrow \forall \epsilon > 0 \exists \delta > 0(0<|x-c|<\delta \implies |F'(c)-F'(x)|<\epsilon) $$
And $F'(x)$ would make sense in a neighbourhood of $c$, then I could use the definition to prove the continuity of $F'(x)$ at $c$.
Thanks for your comments and time!
