A corollary of the fundamental theorem of calculus states:
If $f$ is continuous on $[a, b]$ and $f=g'$ for some $g$, then$$\int_a^b f = g(b) - g(a)$$
A "second fundamental theorem of calculus" replaces the "continuous" for "integrable":
If $f$ is integrable on $[a, b]$ and $f=g'$ for some $g$, then$$\int_a^b f = g(b) - g(a)$$
We also know that if $f$ is differentiable at $a$, then $f$ is continuous at $a$.
I have an issue reconciling the above statement with the second fundamental theorem of calculus. If there is a $g$ such that $f=g'$, then $f$ is a derivative of $g$. Then that says that if $g$ is differentiable, which it should be, then $g$ is continuous. I thought that this implies that $f$ must be continuous. But I guess the fact that $g$ is continuous does not tell us that $g'$ is continuous, and thus $f$ is?