Thomae's function

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Point plot of the function on (0,1)

Thomae's function, also known as the popcorn function, the raindrop function, the ruler function or the Riemann function, is a modification of the Dirichlet function. This real-valued function f(x) is defined as follows:

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

It is assumed here that $gcd(p, q) = 1$ and $q > 0$ so that the function is well-defined and nonnegative ( $gcd$ refers to the greatest common divisor).

1 Discontinuities
2 Follow-up
3 See also
4 External links
5 References

[edit] Discontinuities

The popcorn function is perhaps the simplest example of a function with a complicated set of discontinuities: f 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 $y\ne x$ 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

$\lim_{y\to x} f(y)=0\ne f(x)$

The name "popcorn function" stems from the fact that the graph of this function resembles a snapshot of popcorn popping.^{[citation needed]} It also looks like the interval markers of a ruler or a rainstorm, hence the names "ruler function"^[1] and "raindrop function".^{[citation needed]}

[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; 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 meager 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. If $\textstyle A=\bigcup_{n=1}^{\infty}F_n$ is a countable union of closed sets $F n$ , define

$f_A(x)=\begin{cases}\frac{1}{n}\mbox{ if }x\mbox{ is rational and }n\mbox{ is minimal so that }x\in F_n\\ \\ \frac{-1}{n}\mbox{ if }x\mbox{ is irrational and }n\mbox{ is minimal so that }x\in F_n\\ \\ 0\mbox{ if }x\notin A\end{cases}$

Then a similar argument as for the popcorn function shows that $f A$ has A as its set of discontinuities.

[edit] See also

Euclid's orchard

[edit] External links

Dirichlet Function at MathWorld http://mathworld.wolfram.com/DirichletFunction.html

[edit] References

^ "...the so-called "ruler function", a simple but provocative example that appeared in a work of Johannes Karl Thomae ... The graph suggests the vertical markings on a ruler - hence the name." William Dunham, The Calculus Gallery, Chapter 10

Robert G. Bartle and Donald R. Sherbert (1999), Introduction to Real Analysis, 3rd Edition (Example 5.1.6 (h)). Wiley. ISBN 978-0471321484
Abbot, Stephen. Understanding Analysis. Berlin: Springer, 2001. ISBN 0-387-95060-5