Proof
We will assume that \(F(x)\) is a continuous function. With that assumption, note
\begin{equation*}
P(a-\epsilon \lt x \le a) = \int_{a-\epsilon}^a f(x) dx = F(a) - F(a-\epsilon)
\end{equation*}
Take the limit as \(\epsilon \rightarrow 0^+\) to get the result.