So I'm following a book on complex analysis, and recently got to a theorem that states the following:
If $C$ is a smooth curve, $f$ is a function defined on $C$, and there exists some function $F$, analytic on $C$ such that $F' = f$, then $\int_C f = F(b) - F(a)$ where $a$ and $b$ are, respectively, the beginning and endpoints of the curve.
My question is: would this still be true if, instead of considering $F$ analytic, we consider it just differentiable in $C$? I don't see any reason why the proof would fail.