Exercise 2, § 3

Let $f$ and $g$ be two real regulated functions on a compact interval $[a, b]$, such that $f$ is decreasing on $[a, b]$, and $0 \leq g(t) \leq 1$. If one puts $\lambda = \int_a^b g(t) \, dt$, show that one has

except when $f$ is constant, or $g$ is equal to 0 (resp. 1) at all the points where it is continuous (in which case all three terms are equal). (Vary one of the limits of integration in the integral $\int_1^x f(t)g(t)\,dt$.)