Interactive Real Analysis
- part of Example 7.4.2(a): Simple Functions
A step function is a function s(x) such that
s(x) = cj for
xj-1 < x < xj
and the
{ xj } form a partition of [a, b].
Upper, Lower, and Riemann sums are examples of step functions.
What is the difference, if any, between step functions and simple
functions.
Context
Suppose
s(x) = cj for
xj-1 < x < xj
and the
{ xj } form a partition of [a, b].
Define functions Xj(x) such that
Xj = 1 if
xj-1 < x < xj and 0
otherwise. Then each Xj is a characteristic function
of the interval [xj-1, xj] and the sum
ContextS(x) =is a simple function because (sub)intervals are measurable. But S(x) = s(x), so that every step function is also a simple function.cj Xj(x)
But not every simple function is a step function. Take, for example, the set Q of rational numbers inside [0, 1] and A = [2, 3]. Then the function
S(x) = XQ(x) + XA(x)is a simple function but not a step function.
Therefore, simple functions are more general than step functions.