Let $U$ be uniformly distributed on $[0, 1]$. Denote by $\Phi$ the distribution function of the standard Gaussian distribution. Then $Z = \Phi^{-1}(U)$ has the standard Gaussian distribution.
The domain of $U$ is $[0, 1]$, and the domain of Gaussians is the real line $(-\infty, \infty)$. However, the domain of $Z$ seems to be the extended real line $[-\infty, \infty]$.
Where is the flaw in the above reasoning?