# Tag Archives: counterexamples

## Integrability and Continuity

I just posted a short essay that defines a function with some curious properties. The function is defined by

$f(x) = \begin{cases}\frac{1}{b} & x = \frac{a}{b}\in\mathbf{Q}\text{ in lowest terms}\\ \\0 & \text{if } x \in \mathbf{R}\setminus\mathbf{Q}.\end{cases}$

In the essay I prove that

1. $$f(x)$$ is continuous on $$\mathbf{R}\setminus\mathbf{Q}$$
2. $$f(x)$$ is discontinuous on $$\mathbf{Q}$$
3. $$f(x)$$ is integrable, and in particular $$\int_0^1 f(x)\, dx = 0$$

At first blush, these properties may seem counterintuitive.

The theory of Lebesgue measure completely characterizes Riemann integrable functions: they are functions which are almost everywhere continuous. The above function is almost everywhere continuous because its set of discontinuities ($$\mathbf{Q}$$) is countably infinite, and therefore has Lebesgue measure $$0$$.