Abel's theorem on convergence of power series

Statement

Consider a power series:

$\sum _{k=0}^{\mathrm{\infty }}{a}_k(x-\mu {)}^k$

Suppose that the power series converges to a function $f(x)$ on the interval $(\mu -c,\mu +c)$, i.e.,:

$f(x)=\sum _{k=0}^{\mathrm{\infty }}{a}_k(x-\mu {)}^k\mathrm{\forall }x\in (\mu -c,\mu +c)$

Then, we have the following:

1. If $f$ is defined and left continuous at $\mu +c$ and the power series converges for $x=\mu +c$, then the value to which the power series converges equals $f(\mu +c)$.
2. If $f$ is defined and right continuous at $\mu -c$ and the power series converges for $x=\mu -c$, then the value to which the power series converges equals $f(\mu -c)$.