Assume \(X\) is continuous and \(f\) and \(F\) as above. Notice, by the definition of \(f\text{,}\) \(\lim_{x \rightarrow \pm \infty} f(x) = 0\) since otherwise the integral over the entire space could not be finite.
Now, let \(A(x)\) be any antiderivative of \(f(x)\text{.}\) Then, by the Fundamental Theorem of Calculus,
\begin{align*}
F(x) & = \int_{-\infty}^x f(u) du\\
& = A(x) - \lim_{u \rightarrow -\infty} A(u)
\end{align*}
Hence, \(F'(x) = A'(x) - \lim_{u \rightarrow -\infty} A'(u) = f(x)\) as desired.