I know that generating function $f(x)$ for the Catalan numbers is \begin{equation} f(x)=\cfrac{1\pm \sqrt{1-4x}}{2x}\ . \end{equation}
It is often said that we should choose \begin{equation} f(x)=\cfrac{1- \sqrt{1-4x}}{2x} \end{equation}
because $f(x)$ should be continuous at $x=0$, but I can't understand why $f(x)$ should be continuous.
What is the problem if $f(x)$ is not continuous at $x=0$ ?