Thomae's function

Thomae's function, named after Carl Johannes Thomae, also known as the popcorn function, the raindrop function, the modified Dirichlet function, the ruler function[1], the Riemann function or the Stars over Babylon (by John Horton Conway) 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}

If x = 0 we take q = 1. It is assumed here that gcd(p, q) = 1 and q > 0 so that the function is well-defined and non-negative.

Contents

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.

Informal Proof

Clearly, f is discontinuous at all rational numbers: since the irrationals are dense in the reals, for any rational x, no matter what ε we select, there is an irrational a even nearer to our x where f(a) = 0 (while f(x) is positive). In other words, f can never get "close" to any positive number because its image is dense with zeroes.

To show continuity at the irrationals, assume without loss of generality that our ε is rational (for any irrational ε, we can choose a smaller rational ε and the proof is transitive). Since ε is rational, it can be expressed in lowest terms as a/b. We want to show that f(x) is continuous when x is irrational.

Note that f takes a maximum value of 1 at each whole integer, so we may limit our examination to the space between \lfloor x \rfloor and \lceil x \rceil. Since ε has a finite denominator of b, the only values for which f may return a value greater than ε are those with a reduced denominator no larger than b. There exist only a finite number of values between two integers with denominator no larger than b, so these can be exhaustively listed. Setting δ to be smaller than the nearest distance from x to one of these values guarantees every value within δ of x has f(x) < ε.

Integrability

The Lebesgue criterion for integrability states that a bounded function is Riemann integrable if and only if the set of all discontinuities has measure zero.[2] Since the set of all discontinuities is the rational numbers, and the rational numbers are countable, the set has measure zero. As well, the function is bounded on the interval [0,1], so by the Lebesgue criterion, the function is Riemann integrable on [0,1].

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 Fσ 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.

See also

Notes

  1. ^ "...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
  2. ^ Spivak, M. (p.53, Theorem 3-8)

References

  • Robert G. Bartle and Donald R. Sherbert (1999), Introduction to Real Analysis, 3rd Edition (Example 5.1.6 (h)). Wiley. ISBN 978-0471321484
  • Spivak, M. Calculus on manifolds. 1965. Perseus Books. ISBN 0-8053-9021-9
  • Abbot, Stephen. Understanding Analysis. Berlin: Springer, 2001. ISBN 0-387-95060-5

External links