Functions integrable and not

The theory of integration that we have discussed was developed rigorously by Bernhard Riemann, and this type of integral is known as the Riemann integral.

As we have seen, the Riemann integral calculates the area under the curve \(y=f(x)\) over the interval \([a,b]\) by approximating it with \(n\) rectangles (or trapezia) of equal width, and in the limit, as \(n \to \infty\), the estimate approaches the true area.

Riemann integration works for any function \(f \colon [a,b] \to \mathbb{R}\) that is continuous — roughly, when the graph \(y=f(x)\) can be drawn without taking your pen off the paper. It also works for any piecewise continuous function — where you may take your pen off the paper, but only a finite number of times.

A function \(f(x)\) where the area estimates (based on \(n\) rectangles or trapezia) approach the true integral as \(n \to \infty\) is called Riemann integrable. The previous paragraph, then, says that any piecewise continuous function is integrable.3

An example of a non-integrable function, consider the following rather strange function \(f \colon \mathbb{R} \to \mathbb{R}\),

\[ f(x) = \begin{cases} 1 &\text{if \(x\) is rational}\\ 0 & \text{if \(x\) is irrational.} \end{cases} \]

This function is discontinuous. In fact, near any real number \(x\), there is a rational number arbitrarily close, and there is an irrational number arbitrarily close. So \(f(x)\) is discontinuous at every real number \(x\).

This example of a non-integrable function is quite pathological, and unlikely to arise in real-world situations. There is, however, a theory of integration that can integrate even functions like these. It is called Lebesgue integration and it is developed in university mathematics courses.

Next page - Answers to exercises