If a $f\in\mathbb{C}[[t]]$ is integral over $\mathbb{C}[t]$, convergent for $|t| < r$, then does it extend to a continuous function on $|t|\le r$?

Question

Let $f = \sum_n c_nt^n \in\mathbb{C}[[t]]$ be a power series, and suppose it is integral over the subring $\mathbb{C}[t]$ (ie,it satisfies a monic polynomial with coefficients in $\mathbb{C}[t]$). Suppose $f$ has a positive radius of convergence $r$. Must $f$ extend to a continuous function on the closed disk $\{t\in\mathbb{C} : |t|\le r\}$?