I know that
$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$$ is convergent if Re(s)>1.
And I read that it can be defined continuously for $s\in \mathbb C$ if $1\neq Re(s)>0$. How can this be true?
If $s=\frac{1}{2}$, then $\zeta(\frac{1}{2})=\infty$, just as $\zeta(1)$. So why we can redefine $\zeta$ at $\frac{1}{2}$ to make it continuous but not at $1$?