Thomae's function

If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits.

The popcorn function is the real-valued function f(x) defined as follows

$f(x)={\begin{cases}{\frac {1}{q}}{\mbox{ if }}x={\frac {p}{q}}{\mbox{ a rational number}}\\0{\mbox{ if }}x{\mbox{ is irrational}}\end{cases}}$

Here it is assumed that rational numbers are written in normalized form ${\frac {p}{q}}$ with p, q having no common prime factors and with $q>0$.

The popcorn function is perhaps the simplest example of a function with a complicated set of discontinuities: f(x) is continuous at all irrational numbers and discontinuous at all rational numbers. This may be seen informally as follows: if x is irrational, and y is very close to x, then either y is also irrational, or y is a rational number with a large denominator. Either way, f(y) is close to f(x)=0. On the other hand, if x is rational and $y\neq x$ is very close to x, then it is also true that either y is irrational, or y is a rational number with a large denominator. Thus it follows that

$\lim _{y\to x}f(y)=0\neq f(x)$

The monicker "popcorn function" stems from the fact that the graph of this function resembles a snapshot of popcorn popping.