Complex antiderivative

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 $z=2{e}^{i\theta}$ positively oriented with $-\pi \le \theta \le \pi $? That is, because ${F}^{\prime}(z)=1/z$ is undefined at 0, it is disqualified (I think). But why?

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 $z=2{e}^{i\theta}$ positively oriented with $-\pi \le \theta \le \pi $? That is, because ${F}^{\prime}(z)=1/z$ is undefined at 0, it is disqualified (I think). But why?