6 1. The Regulated and Riemann Integrals

Proof. Suppose that there is another “integral” defined on step func-

tions and satisfying I–V. We will denote this alternate integral as

b

a

f(x) dx.

What we must show is that for every step function f(x),

b

a

f(x) dx =

b

a

f(x) dx.

Suppose that f has partition x0 = a x1 ··· xn−1 xn = b

and satisfies f(x) = ci for xi−1 x xi.

Then, from the additivity property,

(1.3.1)

b

a

f(x) dx =

n

i=1

xi

xi−1

f(x) dx.

But on the interval [xi−1, xi] the function f(x) is equal to the constant

function with value ci except at the endpoints. Since functions which

are equal except at a finite set of points have the same integral, the

integral of f is the same as the integral of ci on [xi−1, xi]. Combining

this with the constant function property we get

xi

xi−1

f(x) dx =

xi

xi−1

ci dx = ci(xi − xi−1).

If we plug this value into equation (1.3.1) we obtain

b

a

f(x) dx =

n

i=1

ci(xi − xi−1) =

b

a

f(x) dx.

Exercise 1.3.4.

(1) Prove that the collection of all step functions on a closed

interval [a, b] is a vector space of functions which contains

the constant functions.

(2) Prove that if x0 = a x1 x2 ··· xn−1 xn = b is a

partition for a step function f with value ci on (xi−1, xi) and