I have given the following question: Argue that $F$ is continuous on $\mathbb{C}$. $F$ is given by $$ F(z)=\sum_{n=0}^\infty \frac{(-1)^n2^{-n}z^{2n+1}}{n!} $$ In a previous exercise we have found that the convergence of radius $R=\infty$. To argue that $F$ is continuous i have used the following sentence: Let $R\in[0,\infty]$ be the convergence of radius for the power series: $\sum_{n=0}^\infty a_n(z-a)^n$. Then the power series' sumfunction is continuous in $\{w\in\mathbb{C}: |w-a|<R\}$
Is it enough to argue that because $R\in[0,\infty]$ then $F$ is continuous as the sentence then is aplied.
Yes, when $R=\infty$, $\{w:|w-a|<R\}$ is the entire complex plane. Hence there is continuity at all points.