Power series are continuous

Let \(f(z) = \lim_{N\to\infty} S_N(z)\) with \(S_N(z)=\sum_{k\leq N} c_k (z-z_0)^k\), and let \(R>0\) be the radius of convergence. The domain of convergence is the open disk \(\abs{z-z_0}<R\), the interior of the set where the series converges. For \(0<r<R\), the series converges absolutely and uniformly on the closed disk \(\abs{z-z_0}\le r\) (Weierstrass \(M\)-test with \(M_k=\abs{c_k}r^k\), which is summable since \(r<R\)). Each partial sum \(S_N\) is a polynomial, hence continuous, so \(f\) is continuous on \(\abs{z-z_0}\le r\) by the uniform limit theorem. Every point of the open disk \(\abs{z-z_0}<R\) lies in such a closed disk, so \(f\) is continuous on the open disk of convergence.