In this edition of the muddiest point I will go over non-Riemann integrable functions. Let’s review a classic example that we mentioned in class, the Dirichlet function. Define the Dirichlet Function as
Recall that by definition, A bounded function defined on the interval is Riemann-integrable if . For , the set of all possible partitions of , we call the upper integral of . Similarly, is the lower integral of . (Notice that these use the lower and upper sums that we discussed in class. We define the lower sum and the upper sum ).
Well, looking at the Dirichlet function, we can see that , but . It follows that , so the function is not Reimann integrable. This way of showing that the upper and lower integrals of a functions are not equal will most likely be one of our main tools to show that a function is not Riemann integrable.