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
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