- #1

Mark53

- 93

- 0

## Homework Statement

Let f [a, b] → R be a non-decreasing function; that is, f(x1) ≤ f(x2) for any x1, x2 ∈ [a, b] with x1 ≤ x2. So f attains a minimum value of m = f(a) and a maximum value of M = f(b) on [a, b]. Let Pn be a regular partition of [a, b] into n subintervals, each of length ∆x = (b − a)/n, and let mi and Mi be the minimum and maximum values of f on the i-th subinterval respectively for each i = 1, 2, . . . , n.

(a) Explain why Mi = mi+1 for each i = 1, 2, . . . , n − 1.

(b) Hence show that U(f,Pn) − L(f,Pn) = (Mn − m1) ∆x.

(c) Express (Mn−m1) ∆x in terms of f, a, b, n and use this to explain why f is integrable on [a, b].

## The Attempt at a Solution

[/B]

A) this is because Mi must be less than mi as it is a non-decreasing function that is why Mi = mi+1

b)

=Mi∆x-mi∆x

=(Mi-mi)∆x

is this part correct?

c)

(Mn−m1) ∆x

(f(a)-f(b))((b-a)/n)

how do i show that it is integrable from here?