Power Series Converges to Continuous Function

Theorem

Let $\displaystyle \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$.

Let $R$ be the radius of convergence of $S$.

Then $\map f x$ is a continuous function on $\set {x: \size {x - \xi} < R}$.

Proof

Let $\rho \in \R$ such that $0 \le \rho < R$.

From Power Series Converges Uniformly within Radius of Convergence, $\map f x$ is uniformly convergent on $\set {x: \size {x - \xi} \le \rho}$.

From Real Polynomial Function is Continuous, each of $\map {f_n} x = a_n x^n$ is a continuous function of $x$.

The result follows from Uniformly Convergent Series of Continuous Functions is Continuous.

$\blacksquare$