Thomae's function

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:

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

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 extstyle A=igcup_{n=1}^{infty}F_n is a countable union of closed sets F_n, define

:f_A(x)=egin{cases}frac{1}{n}mbox{ if }xmbox{ is rational and }nmbox{ is minimal so that }xin F_n\ \frac{-1}{n}mbox{ if }xmbox{ is irrational and }nmbox{ is minimal so that }xin F_n\ \0mbox{ if }x otin Aend{cases}

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


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