Let $D \subset \mathbb C$ be a domain (open and connected set) with some properties: $\mathrm{cl}(D)$ is compact, $\mathrm{int}(\mathrm{cl}(D)) = D$, $\partial D = \coprod_{k = 1}^n \gamma_k$, where $\gamma_k$ is piecewise-smooth simple closed curve and $D$ is always lies left-hand side of $\gamma_k$. $f: \mathbb C \rightarrow \mathbb C$ be differentiable on $D$ and continuous on $\partial D$. Is it true that $\int_{\partial D} f(z) dz = 0$?
I know that this fact is true if $D$ is simply connected (by Cauchy's theorem). In our analysis course the professor said that this fact is true in more general way and to prove it we need to partition $D$ over simply connected pieces and apply Cauchy's theorem on them.
My efforts to prove this theorem:
Let $\epsilon(x) = \sup \{c > 0: f|_{B(x, c) \cap D} \text{ has a derivative}\}$. Function $\epsilon$ is continuous on $\textrm{cl}(D)$ (didn't prove it but it seems to be true), since that there is $\epsilon_0 = \frac {\min\{\text{minimal diameter of "hole" in }D, \min_{x \in \mathrm{cl}(D)} \epsilon(x)\}} 4$. Let consider the $\epsilon_0$-lattice on complex plane. Then I want to show that for each closed square $\Delta$ of this lattice there are finite amount of components of $\Delta \cap \mathrm{cl}(D)$ and each one is simply connected. Then I want to show that the sum of boundaries of components coincides with $\partial D$. However, I'm stuck in technical details of providing this construction.
Can you give me some references or tips which can be helpful in proving this statement?
Edit: By minimal diameter of hole in $D$ I mean $\min_{C \textrm{ is a component of }\mathbb C \setminus \mathrm{cl}(D)} \mathrm{diam}(C)$