Product of "Gaussian" integrals

Calculus Level 5

\[ \large f(z) = \left( \frac{z}{z-1} \right)^{2} \left( \int_{0}^{\infty} e^{-x^{z}} \ dx \right) \left( \int_{0}^{\infty} e^{-y^{z/(z-1)}} \ dy \right)\]

has infinitely many singularities over the real line.

Find the largest value of \(z\) such that there is a singularity at \(z\) and

\[ \displaystyle \lim_{z_{+} \rightarrow z^{+}} f(z_{+}) \neq \lim_{z_{-} \rightarrow z^{-}} f(z_{-})\]

Image Credit: Wikimedia Gaussian Integral
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