On my complex analysis course we have the following statement (Cauchy's integral theorem for star domains): Let $G\subset\mathbb{C}$ be a star domain and let $f:G\to\mathbb{C}$ be a continuous function. Let $f$ be on $G$ holomorphic except for a finite number of points. Then for every closed curve $\alpha$ in $G$ we have \begin{align*} \int_\alpha f(z)\ dz=0. \end{align*}
Then we have this lemma:
Let $R>0$ und $f:D_R(z_0)\to\mathbb{C}$ holomorphic outside the point $z_1\in D_R(z_0),z_1\neq z_0$. If $0<r<R$ and $\varepsilon>0$ are chosen, so that $D_\varepsilon(z_1)\subset D_r(z_0)$, then we have \begin{align*} \int_{\partial D_r(z_0)} f(z)\ dz=\int_{\partial D_\varepsilon(z_1)} f(z)\ dz. \end{align*}
My question is the following: the curve $\partial D_r(z_0)\subset D_R(z_0)$ is closed, the fuction $f$ is on $D_R(z_0)\setminus\{z_1\}$ holomorphic (and specially continuous) and the set $D_R(z_0)$ is a star domain. Due to the Cauchy's integral theorem for star domains above, should be following statement true? \begin{align*} \int_{\partial D_r(z_0)} f(z)\ dz=0. \end{align*}