Step functions are functions , that are constant on sub-intervals of [a,b]; they are also called piecewise constant functions. In order to give a precise definition of this kind of functions it's useful to introduce the concept of dissection (or subdivision) of an interval of real numbers.
A dissection of an interval [a,b] of real numbers is a set of n+1 points such that: . We'll use the traditional set notation for dissections: .
Given a dissection of [a,b], the interval [a,b] can be written as , that is as a finite union of subintervals: these subintervals are generated by the dissection.
A function is called a
step function on [a,b] if
there exists a dissection D of [a,b] and real
numbers
c_{1}, c_{2}, ... ,
c_{n}, such that, for every k = 1, 2,
... ,n, .
Observe that we have imposed no condition on s(x_{k}), except that they are real valued. This is in view of the fact that the Riemann integral is unchanged if we alter the value of the function in a finite number of points.
Given two step functions, s and t, and the corresponding two dissections D and E of their common domain [a,b], it's a useful exercise to prove that the two functions s+t and s·t are step functions and a possible dissection of [a,b] is simply DE.