Are there functions that are not Riemann integrable?
Yes there are, and you must beware of assuming that a function is integrable without looking at it. The simplest examples of non-integrable functions are: in the interval [0, b]; and in any interval containing 0. These are intrinsically not integrable, because the area that their integral would represent is infinite. There are others as well, for which integrability fails because the integrand jumps around too much. An extreme example of this is the function that is 1 on any rational number and 0 elsewhere. Thus the area chosen to represent a single slice in a Riemann sum will be either its width or 0 depending upon whether we pick a rational x or not at which to evaluate our integrand in that interval. For this function no matter how small the intervals are, you can have a Riemann sum of 0 or of b – a. In this case it is possible to use a cleverer definition of the area to define it. (You can argue, in essence, that there are so many more irrational points than rational ones, you can
