Generating function for the Catalan numbers 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 $x=0$ ?

We choose the negative sign in
\begin{align*}
f(x)=\frac{1\color{blue}{\pm} \sqrt{1-4x}}{2x}
\end{align*}
since we want to expand $f$ in a power series.

According   to the  binomial series expansion we have for $|x|<\frac{1}{4}$  the following representation at $x=0$
  \begin{align*}
\sqrt{1-4x}&=\sum_{n=0}^\infty \binom{\frac{1}{2}}{n}(-4x)^n\\
&=1-2x-2x^2-4x^3-10x^4-\cdots
\end{align*}
  so  that
  \begin{align*}
\frac{1+\sqrt{1-4x}}{2x}=\color{blue}{\frac{1}{x}}-1-x-2x^2-5x^3-14x^4-\cdots\tag{1}
\end{align*}
  whereas
  \begin{align*}
\frac{1-\sqrt{1-4x}}{2x}=1+x+2x^2+5x^3+14x^4+\cdots\tag{2}
\end{align*}

Note the  latter (2)  is a   power  series,  while  the former (1) is not a  power series.
