I'm trying to understand complex path integral $\int_C f(z)dz$ for continuous closed curve $C$.
Is it necessary that $C$ is rectifiable and not just generally continuous?
Do we get all the topological results with rectifiable curves? Like winding number for continuous and closed curves, and from that Brouwer fixed point theorem.
So what are the conditions when we move from continuously differentiable curves to continuous curves in integral?