## Abstract

As was pointed out in the previous chapter, the second fundamental topic covered in calculus is the *Riemann Integral*, the first being the *derivative*. For a (nonnegative) real-valued function of a real variable, this integral extends the notion of *area*, defined initially for *rectangles:* For a *nonnegative* constant function *f*(*x*) := *c* ∀*x* ∈ [*a*, *b*], the area of the rectangle bounded by the graph of *f*, the *x*-axis, and the vertical lines *x* = *a* and *x* = *b*, is defined to be the non-negative number *A* := (*b* - *a*)*c*. This is then trivially extended to the case of *step functions* which are *piecewise constant* Simply add the areas of the finite number of rectangles involved. This suggests the following *analytic* approach to the general case: Try to approximate the given function by step functions, find the areas corresponding to the latter functions as above, and *pass to the limit.* Our objective in this chapter is to provide a mathematically rigorous foundation for this intuitive approach. We begin with some basic definitions.

## Keywords

Step Function Monotone Function Open Interval Bounded Variation Measure Zero## Preview

Unable to display preview. Download preview PDF.