Exercise 16, § 1

Functions of a Real Variable, II, 82

Let $\mathbf{f}$ be a regulated vector function on $I = [a, b] \subset \mathbf{R}$ , let $\mathbf{u}$ be a primitive of $\mathbf{f}$ on $I$ and $D$ a closed convex set containing $\mathbf{u}(I)$ . Show that if $g$ is a real monotone function on $I$ one has

\int_a^b \mathbf{f}(t) g(t) \, dt = (\mathbf{u}(b) - \mathbf{c}) g(b) + (\mathbf{c} - \mathbf{u}(a)) g(a)

where

\mathbf{c}

belongs to

D

(reduce to the case where

g

is a monotone step function). Deduce that if

f

is a real regulated function on

I

then there exists a

c \in I

such that

\int_a^b f(t) g(t) \, dt = g(a) \int_a^c f(t) \, dt + g(b) \int_c^b f(t) \, dt

("the second mean value theorem").

Solution

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