1.3. Step Functions 5
Definition 1.3.1. (Step function). A function f : [a, b] R is
called a step function provided there are numbers
x0 = a x1 x2 ··· xn−1 xn = b
such that f(x) is constant on each of the open intervals (xi−1, xi).
It is not difficult to see that the collection of all step functions
defined on [a, b] is a vector space of real-valued functions (see part (1)
of Exercise 1.3.4).
We will say that the points x0 = a x1 ··· xn−1 xn = b
define an interval partition for the step function f. Note that the
definition states that on the open intervals (xi−1, xi) of the partition
f has a constant value, say ci, but it says nothing about the values at
the endpoints. The value of f at the points xi−1 and xi may or may
not be equal to ci. Of course when we define the integral this won’t
matter because the endpoints form a finite set.
Since the area under the graph of a positive step function is a
finite union of rectangles, it is fairly obvious what the integral should
be. The
ith
of these rectangles has width (xi xi−1) and height ci
so we should sum up the areas ci(xi xi−1). If some of the ci are
negative then the corresponding ci(xi xi−1) are also negative, but
that is appropriate since the area between the graph and the x-axis
is below the x-axis on the interval (xi−1, xi).
Definition 1.3.2. (Integral of a step function). Suppose f(x) is a
step function with partition x0 = a x1 x2 ··· xn−1 xn = b
and suppose f(x) = ci for xi−1 x xi. Then we define
b
a
f(x) dx =
n
i=1
ci(xi xi−1).
We made the “obvious” definition for the integral of a step func-
tion, but in fact, we had absolutely no other choice if we want the
integral to satisfy properties I–V above.
Theorem 1.3.3. The integral as given in Definition 1.3.2 is the
unique real-valued function defined on step functions which satisfies
properties I–V of §1.2.
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