Continuity at the boundary of a convergent power series

Question

Say $f(z)=\sum a_{n}z^{n}$ is a power series with convergence radius $0<R<\infty$. Suppose we know that the series convergence at $z_{0}$ where $z_{0}$ is a point at the boundary of the convergence disk. Is $f(z)$ neccaserily continuous at $z_{0}$? That is, if $|z_{n}|<R$ and $z_{n}\to z_{0}$, does $f(z_{n})\to f(z_{0})$? I suspect not because I think that the converence may not be uniform near the boundary, but I don't have an example. Any hints or references would be great, thanks!