[Math] If F is differentiable at c, then f is continuous at c

real-analysisriemann-integration

I am having trouble coming up with a counterpoint to this claim.
I did some work on it, but I don't think it actually makes sense. [https://i.imgur.com/k0f3pDs.jpg][1] I'm doing part (c). Are there any minds more capable than mine that can check my work?

EDIT:

Question: Let $f$ be Riemann integrable on $[a,b]$, let $F(x)=\int_a^x f$, (a<=x<=b), and let $c$ be in $(a,b)$. For each of the following statements, give either a proof or a counterexample.

(c) If $F$ is differentiable at $c$, then $f$ is continuous at $c$

Best Answer

Let $F:\mathbb{R} \to \mathbb{R}$ be defined by $F(x) = x^2 \sin\left(\frac{1}{x}\right)$ with $F(0) = 0$.

Let $G(x) := F(x) - F(-1)$. Since $F$ is everywhere differentiable with an integrable derivative, it follows that.

$$G(x) = \int_{-1}^{x} F'$$

We can note that $G$ is differentiable at $0$, but $F'$ is not continuous there.

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