Thomae's function
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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:
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).
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 y≠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
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".
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 Failed to parse (unknown function\textstyle): \textstyle A=\bigcup_{n=1}^{\infty}F_n , is a countable union of closed sets Fn, define
Then a similar argument as for the popcorn function shows that fA has A as its set of discontinuities.
External links
- Dirichlet Function at MathWorld http://mathworld.wolfram.com/DirichletFunction.html
References
- 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
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