Thomae's function

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

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 $mbox\{gcd\}(p,q)=1$ 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" 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 $y\; e\; x$ 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

:$lim\_\{y\; o\; x\}\; f(y)=0\; e\; f(x)$

The name "popcorn function" stems from the fact that the graph of this function resembles a snapshot of popcorn popping.Fact|date=April 2008 It also looks like the interval markers of a ruler or a rainstorm, hence the names "ruler function" ["...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] 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; 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 is a countable union of closed sets $F\_n$, define

:

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

ee also

* Euclid's orchard – Thomae's function can be interpreted as a perspective drawing of Euclid's orchard

External links

*

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