Use Exercise 47(c) to prove:
(a) F'(r) is an increasing function of r.
(b) Income is distributed equally (all households have the same income) if and only if F(r) = r for 0 ≤ r ≤ 1.

Data From Exercise 47(c)

(c) Let 0 ≤ r ≤ 1. A household belongs to the 100rth percentile if its income is greater than or equal to the income of 100r% of all households. Pass to the limit as Δr → 0 in (b) to derive the following interpretation: A household in the 100rth percentile has income F'(r)A. In particular, a household in the 100rth percentile receives more than the national average if F'(r) > 1 and less if F'(r) < 1