Prove that the quantile function F−1 of a general random variable X has the following three properties that are analogous to properties of the c.d.f.:
a. F−1 is a nondecreasing function of p for 0 < p < 1.
b. Let x0 = F−1(p) and x1 = F−1(p). Then x0 equals the greatest lower bound on the set of numbers c such that Pr(X ≤ c) > 0, and x1 equals the least upper bound on the set of numbers d such that Pr(X ≥ d) > 0.
c. F−1 is continuous from the left; that is F−1(p) = F−1(p−) for all 0 < p < 1.