Let $f\,\,$ be a continuous function on $[a,\infty)$ such that $\int_a^\infty f(t)\,dt$ converges. Define the function $F\,$ on $[a,\infty)$ with
$F(x) := -\int_x^\infty f(t)\,dt \qquad\text{for all}\quad x\in[a,\infty).$
Can we somehow deduce from this—using the regular fundamental theorem of calculus—that $F\,\,$ is continuous and differentiable on $(a,\infty)$, and that F\,\,'(x) = f(x) for all $x\in(a,\infty)$? If so, how? Are such “improper” versions of the fundamental theorem to be found in some book out there that I can reference?