 Solution

Use the quantile function $F^{-1}(u)=\inf \{x:F(x)\ge u\}$ for $0<u<1$, so a strictly increasing CDF is not required. For any real $x$, the quantile event $F^{-1}(U)\le x$ agrees, up to endpoint null sets, with $U\le F(x)$. HenceThus $F^{-1}(U)$ has distribution function $F$ and probability density function $f$. This proves inverse transform sampling, including a probability density function with intervals on which the CDF is constant.