Differentiability
In a standard calculus course, we often learn functions that are differentiable, namely functions that are continuous, have no “sharp turns”, and have no vertical tangent line. So, a function like is differentiable but and are not since has a “sharp turn” at 0 and has a vertical tangent line at 0.
To be more precise, a function is said to be differentiable at if exists. One beautiful result of this definition is that differentiability implies continuity.
Theorem 1.1.1. If is differentiable at , then is continuous at .
Proof. Suppose is differentiable at , so exists. Notice that . Therefore, , thus is continuous at . ∎
Now you may wonder, what are the functions that are integrable? Is there even a function that is not integrable? What is the relationship between continuity and integrability? To answer these questions, we need to rigorously define what it means to be integrable just like how we did for differentiability.
Integrability
In a standard calculus course, it is not often discussed whether a function is integrable. We learn about Riemann sums and how the upper sum is an overestimation and how the lower sum is an underestimation of the area under the curve. And to find the exact area under the curve, we make the width of each rectangle infinitesimally small, so that when you add all those rectangles up, you get the exact area under the curve. So, you have probably seen this: where is any number that is in the subinterval.
Now consider where . This function is also known as the Dirichlet function, named after the German mathematician Peter Gustav Lejeune Dirichlet.
Say we want to compute . No matter how we partition , you can find a rational number and an irrational number in each subinterval since and are both dense in . This means that our smallest value in each subinterval is 0 and the largest value in each subinterval is 1. This implies that letting be the smallest value in that subinterval results in , but letting be the largest value in that subinterval results in . We all know that , so… what went wrong?
We need to know for sure that is integrable before we can compute . Before we do that, let’s introduce some notation. Let of be a finite set of points in (we call this a partition of ). The convention is to list these points in increasing order, so an example of in might be . Now, let arbitary . We define (the smallest value of in subinterval) and (the largest value of in subinterval). Then, the lower and upper sum of with the partition is given by and respectively. Now, let be the set of all possible partitions of . So, any of is an element of . We define lower integral of as (greatest out of every partition ). Similarily, we define upper integral of as (smallest out of every partition ). It is not hard to see that if is bounded.
Finally, let us define what it means for a function to be integrable (more specifically, Riemann-integrable).
Definition 1.1.2. A bounded function defined on is Riemann-integrable if .
If is Riemann-integrable, then . One nice aspect of this definition is that continuity implies integrability.
Now using the Definition 1.1.2, let’s show that the Dirichlet function is not Riemann-integrable on .
Proof. Let be a partition of . If with , then becasue is dense in and because is dense in as well. This implies that for every partition , it follows that and . Therefore, and . Since , then is not Riemann-integrable on . ∎
Indeed, you can extend this to show that the Dirichlet function is not Riemann-integrable on .
You can now see that Riemann-integration is not perfect. In addition to poor handling of functions with many discontinuities, it also has a problem with unbounded functions, such as . Now you are wondering, is there a solution to these problems? The answer is yes: Lebesgue-integration. Lebesgue definition of integral uses horizontal slabs that are not necessarily rectangles unlike Riemann’s, and it can compute intergrals for a much wider range of functions including the Dirichlet function.
Learning more about the Lebesgue-integration is left as an exercise to the reader.