If F is differentiable at c, then f is continuous at c

Question

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.stack.imgur.com/FqsMX.jpg 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 $c$

Answer 1

Hint: Consider the term

$$\frac{f(x)-f(x_0)}{x-x_0}(x-x_0)+f(x_0)$$ and calculate the Limit for $x$ tends to $x_0$

Answer 2

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.