I am confused on a couple things:
1.) Why is it that an integral of a complex valued function of a complex variable exists if f(z(t)) is piecewise continuous (and/or piecewise continuous on $\mathbb{C}$) and not continuous, like a real?
2.) Why is it that one cannot make use of an antiderivative to evaluate an integral of a function like $1/z$, on a contour of something like $z=2e^{i\theta}$ positively oriented with $-\pi \le \theta \le \pi$? That is, because $F'(z)=1/z$ is undefined at $0$, it is disqualified (I think). But why?