Real power series/Convergence/Continuous function/Fact

Let

${\displaystyle {}f(x):=\sum _{n=0}^{\infty }c_{n}x^{n}\,}$

be a power series and suppose that there exists some ${\displaystyle {}x_{0}\neq 0}$ such that ${\displaystyle {}\sum _{n=0}^{\infty }c_{n}x_{0}^{n}}$ converges.

Then there exists a positive ${\displaystyle {}R}$

(where ${\displaystyle {}R=\infty }$ is allowed) such that for all ${\displaystyle {}x\in \mathbb {R} }$ fulfilling ${\displaystyle {}\vert {x}\vert <R}$ the series converges absolutely. On such an (open) interval of convergence, the power series ${\displaystyle {}f(x)}$ represents a

continuous function.