= Solution
Use the <quantile function> $F^{-1}(u)=\inf\{x:F(x)\geq u\}$ for $0<u<1$, so a strictly increasing CDF is not required. For any real $x$, the quantile event $F^{-1}(U)\leq x$ agrees, up to endpoint null sets, with $U\leq F(x)$. Hence
$$
\mathbb P(F^{-1}(U)\leq x)=\mathbb P(U\leq F(x))=F(x).
$$
Thus \b[$F^{-1}(U)$ 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.
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