It is claimed on it's Wikipedia page that Euler's function, defined by the infinite product $\prod\_1^\infty(1-q^n)$ for $|q|<1$, cannot be analytically continued outside the unit disc, that is, the unit disc is a natural boundary of the function. Unfortunately no proof is referred to. 


I would like to know how this claim is proved, as it appears to me that it is defined uniquely for $|q|>1$.



Proof that Euler's function cannot be continued beyond the open disc?
If an analytic function $f(q)$ is defined on some open neighborhood of $q=a$, and there exists a sequence $a\_1,a\_2,\dots$ in this neigborhood with $\lim\_{n\to\infty}a\_n=a$ and $f(a\_n)=0$ for all $n$, then the function $f$ must be identical to $0$. Since the zeroes lie dense on the unit circle, $f$ cannot be defined on any open neigborhood of any point on the unit circle. 
So there is no analytic continuation of the function that can ``escape'' from the unit disc.


