= Solution
Write the Laurent expansion on the punctured disc as
$$
f(z)=\sum_{n=-\infty}^{\infty}a_nz^n.
$$
For every integer $k\geq1$, the Laurent coefficient formula and the <Cauchy-Schwarz inequality> give
$$
\begin{aligned}
|a_{-k}|
&=\left|\frac{r^k}{2\pi}
\int_0^{2\pi}f(re^{i\theta})e^{ik\theta}\,d\theta\right|\\
&\leq\frac{r^k}{2\pi}
\left(\int_0^{2\pi}|f(re^{i\theta})|^2d\theta\right)^{1/2}
\left(\int_0^{2\pi}1\,d\theta\right)^{1/2}\\
&\leq\frac{r^k}{\sqrt{2\pi}}.
\end{aligned}
$$
Letting $r\downarrow0$ shows that $a_{-k}=0$. Every coefficient in the principal part vanishes, so the <uniform L2 circle bound for a removable singularity> proves that
$$
\boxed{f\text{ has a removable singularity at }0}.
$$
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