Singularities of a complex function in the punctured disc Let $f$ be a holomorphic funclion in the punctured disc $\\{z: 0<|z|<1\\}$ such that $|f(z)|\leq 100 + |\log|z||$, $\forall z\in\mathbb{C}$. Does the point admit a singularity?

By Riemann's theorem on removable singularities $$ \bigl\lvert \oint_{\gamma_C} f(z) z^n\,dz \bigr\rvert \leq \oint_{\gamma_C} \bigl\lvert f(z) z^n\bigr\rvert\,dz \leq \oint_{\gamma_C} (100 + \mid \log \mid z \mid \mid) \bigl\lvert z^n\bigr\rvert\,dz = \\\ = 2\pi r^{n+1}100 + \int_{0}^{2\pi}r^n \mid \log r \mid d\varphi \\\ \leq 2\pi r^{n+1}+2\pi r^{n-\frac{1}{2}} \rightarrow 0, r\rightarrow 0$$ where $\gamma_C$ is the path of a small circle around $0$.

This shows that at $0$, the function $f$ has a removable singularity.