= Inverse probability content bound for a Lipschitz density
For a strictly positive <probability density function> with <Lipschitz bound> $L$, $p_x(r)=\int_{x-r}^{x+r}f(y)\,dy$ satisfies $|p_x(r)-2rf(x)|\leq Lr^2$. If $0<s<1$ and $s\leq f(x)^2/L$, monotonicity gives $p_x^{-1}(s)\leq s/f(x)$ and hence $|2f(x)p_x^{-1}(s)-s|\leq Ls^2/f(x)^2$. The <Lipschitz density height bound> prevents extending this local argument to every $s\in(0,1)$ by assuming $f(x)\geq\sqrt L$ for a strictly positive density.
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