Proof verification: if $f:[a,b]\to\mathbb{R}$ is integrable and $\lim_{x\to c\in(a,b)}f(x)$ exists, then $F=\int_a^x f$ is differentiable at $c$.
Allow me to shorten your proof for you. Define
$$g(x) = \begin{cases} f(x) & x \neq c \\ L & x = c \end{cases}$$
Then $g : [a, b] \to \mathbb{R}$ is Riemann integrable, as it differs from the Riemann integrable function $f$ in at most 1 place. Moreover, we see that for all $x \in [a, b]$, we have $F(x) = \int\limits_a^x f(t) dt = \int\limits_a^x g(t) dt$, since $f$ and $g$ differ at at most 1 point. Finally, $g$ is continuous at $c$. Therefore, by your local lemma, we have $F'(c) = g(c) = L$.