Let $X$ be a random variable with c.d.f. $F$ and quantile function $F^{-1}$. Assume the following three conditions:
(i) $F^{-1}(p) = c$ for all $p$ in the interval $(p_0, p_1)$,
(ii) either $p_0 = 0$ or $F^{-1} (p_0) < c$, and
(iii) either $p_1 = 1$ or $F^{-1}(p) > c$ for $p > p_1$.
Prove that $Pr(X=c) = p_1 - p_0$.
No idea how to start proving this condition. Any help will be greatly appreciated. Thanks in advance!