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The popcorn function, also known as Thomae's function, Dirichlet's function, the raindrop function, or the ruler function, is the real-valued function f(x) defined as follows:

<math>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}</math>

It is assumed here that <math>\mbox{gcd}(p,q)=1</math> and <math>q>0</math> so that the function is well-defined and nonnegative.

[edit] Discontinuities

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 <math>y\ne x</math> 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

<math>\lim_{y\to x} f(y)=0\ne f(x)</math>

The moniker "popcorn function" stems from the fact that the graph of this function resembles a snapshot of popcorn popping. It also looks like the interval markers of a ruler or a rainstorm, hence the nicknames "ruler function" and "raindrop function".

[edit] Follow-up

A natural followup question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. For it can be shown that the set of discontinuities of any function must be an F-sigma set. If such a function existed, then the irrationals would be F-sigma and hence would also be a meagre set. It would follow that the real numbers, being a union of the irrationals and the rationals (which is evidently meager), would also be a meager set. This would contradict the Baire category theorem.

A variant of the popcorn function can be used to show that any F-sigma subset of the real numbers can be the set of discontinuities of a function. For if <math>\textstyle A=\bigcup_{n=1}^{\infty}F_n</math>, is a countable union of closed sets <math>F_n</math>, define

<math>f_A(x)=\begin{cases}\frac{1}{n}\mbox{ if }x\mbox{ is rational and }n\mbox{ is minimal so that }x\in F_n\\ \\

\frac{-1}{n}\mbox{ if }x\mbox{ is irrational and }n\mbox{ is minimal so that }x\in F_n\\ \\ 0\mbox{ if }x\notin A\end{cases}</math>

Then a similar argument as for the popcorn function shows that <math>f_A</math> has A as its set of discontinuities.he:פונקציית רימן