This morning I realized I have never understood a technical issue about Cauchy's theorem (homotopy form) of complex analysis. To illustrate, let me first give a definition.
(In what follows $\Omega$ will always denote an open subset of the complex plane.)
Definition Let $\gamma, \eta\colon [0, 1]_t \to \Omega$ be two piecewise smooth closed curves (or circuits). We say that $\gamma, \eta$ are $\Omega$-homotopic if there exists a continuous mapping $H \colon [0, 1]_\lambda \times [0, 1]_t \to \Omega$ s.t.
- $H(0, t)=\gamma(t)$ and $H(1, t)=\eta(t), \quad \forall t \in [0, 1]$;
- $H(\lambda, 0)=H(\lambda, 1), \quad \forall \lambda \in [0, 1]$.
Theorem (Cauchy) Let $f\colon \Omega \to \mathbb{C}$ be holomorphic. If $\gamma, \eta$ are $\Omega$-homotopic circuits, then
$\int_{\gamma} f(z)\, dz= \int_{\eta}f(z)\, dz.$
Problem The function $H$ above is only continuous and need not be smooth. So for $0< \lambda < 1$, the closed curve $H(\lambda, \cdot)$ may be pretty much everything (a Peano curve, for example). Does this void the validity of theorem as it is stated above? How can integration be defined over such a pathological object?
The proof of Cauchy's theorem that I have in mind goes as follows. To begin, one observes that for a sufficiently small value of $\lambda_1$, the circuits $\gamma=H(0, \cdot)$ and $H(\lambda_1, \cdot)$ are close toghether; that is, they can be covered by a finite sequence of disks not leaving $\Omega$ like in the following figure:
Since $f$ is locally exact, its integrals over every single disk depend only on the local primitive. Playing a bit with this, one arrives at
$\int_\gamma f(z)\, dz= \int_{H(\lambda_1, \cdot)} f(z)\, dz.$
Then one repeats this process, yielding a $\lambda_2$ greater than $\lambda_1$ and such that
$\int_{H(\lambda_1,\cdot)} f(z)\, dz= \int_{H(\lambda_2, \cdot)} f(z)\, dz.$
And so on. A compactness argument finally shows that this algorithm ends in a finite number of steps.
Problem is: this proof assumes implicitly that $H(\lambda_1, \cdot), H(\lambda_2, \cdot) \ldots$ are piecewise smooth, to make sense of integrals $\int_{H(\lambda_j, \cdot)}f(z)\, dz.$ This, however, does not follow from the definition if $H$ is only assumed to be continuous. Therefore this proof works only for smooth $H$.
Is this regularity condition necessary?