# Step functions/Integral from above and from below/Section

Let be a real interval with endpoints . Then a function

is called a *step function*, if there exists a partition

of such that is constant

on every open interval .This definition does not require a certain value at the partition points. We call the interval the -th interval of the partition, and is called the length of this interval. If the lengths of all intervals are constant, then the partition is called an *equidistant partition*.

Let be a real interval with endpoints , and let

denote a step function for the partition , with the values , . Then

*step integral*of on .

We denote the step integral also by . If we have an equidistant partition of interval length , then the step integral equals . The step integral does not depend on the partition chosen. As long as we have a step function with respect to the partition, one can pass to a refinement of the partition.

Let denote a bounded interval, and let

denote a function. Then a step function

is called a *step function from above* for , if
holds for all
.
A step function

is called a *step function from below* for , if
holds for all

A step function from above (below) for exists if and only if is bounded from above (from below).

Let denote a bounded interval, and let

denote a function. For a step function from above

of , with respect to the partition , , and values , , the step integral

*step integral from above*for on .

Let denote a bounded interval, and let

denote a function. For a step function from below

of , with respect to the partition , , and values , , the step integral

*step integral from below*for on .

Different step functions from above yield different step integrals from above.

Let denote a bounded interval, and let

denote a function, which is bounded from above. Then the infimum of all step integrals of step functions from above

of is called the*upper integral*of .

Let denote a bounded interval, and let

denote a function, which is bounded from below. Then the supremum of all step integrals of step functions from below

of is called the*lower integral*of .

The boundedness from below makes sure that there exists at all a step function from below, so that the set of step integrals from below is not empty. This condition alone does not guarantee that a supremum exists. However, if the function is bounded from both sides, then the upper integral and the lower integral exist. If a partition is given, then there exists a smallest step function from above (a largest from below) which is given by the suprema (infima) of the function on the intervals of the partition. For a continuous function on a closed interval, these are maxima and minima. To compute the integral, we have to look at all step functions for all partitions.