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 $|q|>1$.