 Solution

The generalized quantile function isThe tail limits of a cumulative distribution function make the defining set nonempty and its infimum finite. Monotonicity and right continuity imply $F^{-1}(u)\le x$ exactly when $u\le F(x)$. To see the direction involving the infimum, let points in the defining set decrease towards it; right continuity gives $F(F^{-1}(u))\ge u$, so every larger $x$ also qualifies. Consequently, for a uniform distribution variable $U$,The probability-zero endpoints $U=0,1$ can be assigned arbitrary outputs. This proves inverse transform sampling even for distributions with atoms or flat portions; continuity or strict monotonicity of $F$ is not required.