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?
Best Answer
$$F(x) := -\int_x^\infty f(t)\,dt = -\int_x^c f(t)\;dt + \int_c^\infty f(t)\;dt$$
The derivative of the first term of this sum with respect to $x$ is $f(x)$, and that of the second term is $0.$ You have some leeway as to what $c$ is; you could just choose $a.$