Thomae's function

Thomae's function is a real-valued function of a real variable that can be defined as:^[1]

Point plot on the interval (0,1). The topmost point in the middle shows f(1/2) = 1/2

${\displaystyle f(x)={\begin{cases}{\frac {1}{q}}&{\text{if }}x={\tfrac {p}{q}}\quad (x{\text{ is rational), with }}p\in \mathbb {Z} {\text{ and }}q\in \mathbb {N} {\text{ coprime}}\\0&{\text{if }}x{\text{ is irrational.}}\end{cases}}}$

It is named after Carl Johannes Thomae, but has many other names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function,^[2] the Riemann function, or the Stars over Babylon (John Horton Conway's name).^[3] Thomae mentioned it as an example for an integrable function with infinitely many discontinuities in an early textbook on Riemann's notion of integration.^[4]

Since every rational number has a unique representation with coprime (also termed relatively prime) ${\displaystyle p\in \mathbb {Z} }$ and ${\displaystyle q\in \mathbb {N} }$, the function is well-defined. Note that ${\displaystyle q=+1}$ is the only number in ${\displaystyle \mathbb {N} }$ that is coprime to ${\displaystyle p=0.}$

It is a modification of the Dirichlet function, which is 1 at rational numbers and 0 elsewhere.

Properties

• Thomae's function ${\displaystyle f}$  is bounded and maps all real numbers to the unit interval:${\displaystyle \;f:\mathbb {R} \;\rightarrow \;[0,\;1].}$ 
• ${\displaystyle f}$  is periodic with period ${\displaystyle 1:\;f(x+n)=f(x)}$  for all integers n and all real x.

Proof of periodicity

For all ${\displaystyle x\in \mathbb {R} \smallsetminus \mathbb {Q} ,}$  we also have ${\displaystyle x+n\in \mathbb {R} \smallsetminus \mathbb {Q} }$  and hence ${\displaystyle f(x+n)=f(x)=0,}$  For all ${\displaystyle x\in \mathbb {Q} ,\;}$  there exist ${\displaystyle p\in \mathbb {Z} }$  and ${\displaystyle q\in \mathbb {N} }$  such that ${\displaystyle \;x=p/q,\;}$  and ${\displaystyle \gcd(p,\;q)=1.}$  Consider ${\displaystyle x+n=(p+nq)/q}$ . If ${\displaystyle d}$  divides ${\displaystyle p}$  and ${\displaystyle q}$ , it divides ${\displaystyle p+nq}$  and ${\displaystyle q}$ . Conversely, if ${\displaystyle d}$  divides ${\displaystyle p+nq}$  and ${\displaystyle q}$ , it divides ${\displaystyle (p+nq)-nq=p}$  and ${\displaystyle q}$ . So ${\displaystyle \gcd(p+nq,q)=\gcd(p,q)=1}$ , and ${\displaystyle f(x+n)=1/q=f(x)}$ .

• ${\displaystyle f}$  is discontinuous at all rational numbers, dense within the real numbers.

Proof of discontinuity at rational numbers

Let ${\displaystyle x_{0}=p/q}$  be an arbitrary rational number, with ${\displaystyle \;p\in \mathbb {Z} ,\;q\in \mathbb {N} ,}$  and ${\displaystyle p}$  and ${\displaystyle q}$  coprime. This establishes ${\displaystyle f(x_{0})=1/q.}$  Let ${\displaystyle \;\alpha \in \mathbb {R} \smallsetminus \mathbb {Q} \;}$  be any irrational number and define ${\displaystyle x_{n}=x_{0}+{\frac {\alpha }{n}}}$  for all ${\displaystyle n\in \mathbb {N} .}$  These ${\displaystyle x_{n}}$  are all irrational, and so ${\displaystyle f(x_{n})=0}$  for all ${\displaystyle n\in \mathbb {N} .}$  This implies ${\displaystyle |x_{0}-x_{n}|={\frac {\alpha }{n}},\quad }$  and ${\displaystyle \quad |f(x_{0})-f(x_{n})|={\frac {1}{q}}.}$  Let ${\displaystyle \;\varepsilon =1/q\;}$ , and given ${\displaystyle \delta >0}$  let ${\displaystyle n=1+\left\lceil {\frac {\alpha }{\delta }}\right\rceil .}$  For the corresponding ${\displaystyle \;x_{n}}$  we have ${\displaystyle |f(x_{0})-f(x_{n})|=1/q\geq \varepsilon \quad }$  and ${\displaystyle |x_{0}-x_{n}|={\frac {\alpha }{n}}={\frac {\alpha }{1+\left\lceil {\frac {\alpha }{\delta }}\right\rceil }}<{\frac {\alpha }{\left\lceil {\frac {\alpha }{\delta }}\right\rceil }}\leq \delta ,}$  which is exactly the definition of discontinuity of ${\displaystyle f}$  at ${\displaystyle x_{0}}$ .

• ${\displaystyle f}$  is continuous at all irrational numbers, also dense within the real numbers.

Proof of continuity at irrational arguments

Since ${\displaystyle f}$  is periodic with period ${\displaystyle 1}$  and ${\displaystyle 0\in \mathbb {Q} ,}$  it suffices to check all irrational points in ${\displaystyle I=(0,\;1).\;}$  Assume now ${\displaystyle \varepsilon >0,\;i\in \mathbb {N} }$  and ${\displaystyle x_{0}\in I\smallsetminus \mathbb {Q} .}$  According to the Archimedean property of the reals, there exists ${\displaystyle r\in \mathbb {N} }$  with ${\displaystyle 1/r<\varepsilon ,}$  and there exist ${\displaystyle \;k_{i}\in \mathbb {N} ,}$  such that for ${\displaystyle i=1,\ldots ,r}$  we have ${\displaystyle 0<{\frac {k_{i}}{i}}<x_{0}<{\frac {k_{i}+1}{i}}.}$  The minimal distance of ${\displaystyle x_{0}}$  to its i-th lower and upper bounds equals ${\displaystyle d_{i}:=\min \left\{\left|x_{0}-{\frac {k_{i}}{i}}\right|,\;\left|x_{0}-{\frac {k_{i}+1}{i}}\right|\right\}.}$  We define ${\displaystyle \delta }$  as the minimum of all the finitely many ${\displaystyle d_{i}.}$  ${\displaystyle \delta :=\min _{1\leq i\leq r}\{d_{i}\},\;}$  so that for all ${\displaystyle i=1,...,r,}$  ${\displaystyle \quad |x_{0}-k_{i}/i|\geq \delta \quad }$  and ${\displaystyle \quad |x_{0}-(k_{i}+1)/i|\geq \delta .}$  This is to say, all these rational numbers ${\displaystyle k_{i}/i,\;(k_{i}+1)/i,\;}$  are outside the ${\displaystyle \delta }$ -neighborhood of ${\displaystyle x_{0}.}$  Now let ${\displaystyle x\in \mathbb {Q} \cap (x_{0}-\delta ,x_{0}+\delta )}$  with the unique representation ${\displaystyle x=p/q}$  where ${\displaystyle p,q\in \mathbb {N} }$  are coprime. Then, necessarily, ${\displaystyle q>r,\;}$  and therefore, ${\displaystyle f(x)=1/q<1/r<\varepsilon .}$  Likewise, for all irrational ${\displaystyle x\in I,\;f(x)=0=f(x_{0}),\;}$  and thus, if ${\displaystyle \varepsilon >0}$  then any choice of (sufficiently small) ${\displaystyle \delta >0}$  gives ${\displaystyle |x-x_{0}|<\delta \implies |f(x_{0})-f(x)|=f(x)<\varepsilon .}$  Therefore, ${\displaystyle f}$  is continuous on ${\displaystyle \mathbb {R} \smallsetminus \mathbb {Q} .\quad }$

• ${\displaystyle f}$  is nowhere differentiable.

Proof of being nowhere differentiable

For rational numbers, this follows from non-continuity. For irrational numbers: For any sequence of irrational numbers ${\displaystyle (a_{n})_{n=1}^{\infty }}$  with ${\displaystyle a_{n}\neq x_{0}}$  for all ${\displaystyle n\in \mathbb {N} _{+}}$  that converges to the irrational point ${\displaystyle x_{0},\;}$  the sequence ${\displaystyle (f(a_{n}))_{n=1}^{\infty }}$  is identically ${\displaystyle 0,\;}$  and so ${\displaystyle \lim _{n\to \infty }\left|{\frac {f(a_{n})-f(x_{0})}{a_{n}-x_{0}}}\right|=0.}$  According to Hurwitz's theorem, there also exists a sequence of rational numbers ${\displaystyle (b_{n})_{n=1}^{\infty }=(k_{n}/n)_{n=1}^{\infty },\;}$  converging to ${\displaystyle x_{0},\;}$  with ${\displaystyle k_{n}\in \mathbb {Z} }$  and ${\displaystyle n\in \mathbb {N} }$  coprime and ${\displaystyle |k_{n}/n-x_{0}|<{\frac {1}{{\sqrt {5}}\cdot n^{2}}}.\;}$  Thus for all ${\displaystyle n,}$  ${\displaystyle \left|{\frac {f(b_{n})-f(x_{0})}{b_{n}-x_{0}}}\right|>{\frac {1/n-0}{1/({\sqrt {5}}\cdot n^{2})}}={\sqrt {5}}\cdot n\neq 0\;}$  and so ${\displaystyle f}$  is not differentiable at all irrational ${\displaystyle x_{0}.}$

• ${\displaystyle f}$  has a strict local maximum at each rational number.^{[citation needed]}

See the proofs for continuity and discontinuity above for the construction of appropriate neighbourhoods, where ${\displaystyle f}$  has maxima.

• ${\displaystyle f}$  is Riemann integrable on any interval and the integral evaluates to ${\displaystyle 0}$  over any set.

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.^[5] Every countable subset of the real numbers - such as the rational numbers - has measure zero, so the above discussion shows that Thomae's function is Riemann integrable on any interval. The function's integral is equal to ${\displaystyle 0}$  over any set because the function is equal to zero almost everywhere.

• If ${\displaystyle G=\{\,(x,f(x)):x\in (0,1)\,\}\subset \mathbb {R} ^{2}}$  is the graph of the restriction of ${\displaystyle f}$  to ${\displaystyle (0,1)}$ , then the box-counting dimension of ${\displaystyle G}$  is ${\displaystyle 4/3}$ .^[6]

Related probability distributions

Empirical probability distributions related to Thomae's function appear in DNA sequencing.^[7] The human genome is diploid, having two strands per chromosome. When sequenced, small pieces ("reads") are generated: for each spot on the genome, an integer number of reads overlap with it. Their ratio is a rational number, and typically distributed similarly to Thomae's function.

If pairs of positive integers ${\displaystyle m,n}$  are sampled from a distribution ${\displaystyle f(n,m)}$  and used to generate ratios ${\displaystyle q=n/(n+m)}$ , this gives rise to a distribution ${\displaystyle g(q)}$  on the rational numbers. If the integers are independent the distribution can be viewed as a convolution over the rational numbers, ${\textstyle g(a/(a+b))=\sum _{t=1}^{\infty }f(ta)f(tb)}$ . Closed form solutions exist for power-law distributions with a cut-off. If ${\displaystyle f(k)=k^{-\alpha }e^{-\beta k}/\mathrm {Li} _{\alpha }(e^{-\beta })}$  (where ${\displaystyle \mathrm {Li} _{\alpha }}$  is the polylogarithm function) then ${\displaystyle g(a/(a+b))=(ab)^{-\alpha }\mathrm {Li} _{2\alpha }(e^{-(a+b)\beta })/\mathrm {Li} _{\alpha }^{2}(e^{-\beta })}$ . In the case of uniform distributions on the set ${\displaystyle \{1,2,\ldots ,L\}}$  ${\displaystyle g(a/(a+b))=(1/L^{2})\lfloor L/\max(a,b)\rfloor }$ , which is very similar to Thomae's function.^[7]

The ruler function

Main article: Ruler function

For integers, the exponent of the highest power of 2 dividing ${\displaystyle n}$  gives 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, ... (sequence A007814 in the OEIS). If 1 is added, or if the 0s are removed, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, ... (sequence A001511 in the OEIS). The values resemble tick-marks on a 1/16th graduated ruler, hence the name. These values correspond to the restriction of the Thomae function to the dyadic rationals: those rational numbers whose denominators are powers of 2.

Related functions

A natural follow-up 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_σ set. If such a function existed, then the irrationals would be an F_σ set. The irrationals would then be the countable union of closed sets ${\displaystyle \textstyle \bigcup _{i=0}^{\infty }C_{i}}$ , but since the irrationals do not contain an interval, neither can any of the ${\displaystyle C_{i}}$ . Therefore, each of the ${\displaystyle C_{i}}$  would be nowhere dense, and the irrationals would be a meager set. It would follow that the real numbers, being the union of the irrationals and the rationals (which, as a countable set, is evidently meager), would also be a meager set. This would contradict the Baire category theorem: because the reals form a complete metric space, they form a Baire space, which cannot be meager in itself.

A variant of Thomae's function can be used to show that any F_σ subset of the real numbers can be the set of discontinuities of a function. If ${\displaystyle A=\textstyle \bigcup _{n=1}^{\infty }F_{n}}$  is a countable union of closed sets ${\displaystyle F_{n}}$ , define

${\displaystyle f_{A}(x)={\begin{cases}{\frac {1}{n}}&{\text{if }}x{\text{ is rational and }}n{\text{ is minimal so that }}x\in F_{n}\\-{\frac {1}{n}}&{\text{if }}x{\text{ is irrational and }}n{\text{ is minimal so that }}x\in F_{n}\\0&{\text{if }}x\notin A\end{cases}}}$ 

Then a similar argument as for Thomae's function shows that ${\displaystyle f_{A}}$  has A as its set of discontinuities.

See also

• Blumberg theorem
• Cantor function
• Dirichlet function
• Euclid's orchard – Thomae's function can be interpreted as a perspective drawing of Euclid's orchard
• Volterra's function

Notes

1. Beanland, Roberts & Stevenson 2009, p. 531
2. "…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." (Dunham 2008, p. 149, chapter 10)
3. John Conway. "Topic: Provenance of a function". The Math Forum. Archived from the original on 13 June 2018.
4. Thomae 1875, p. 14, §20
5. Spivak 1965, p. 53, Theorem 3-8
6. Chen, Haipeng; Fraser, Jonathan M.; Yu, Han (2022). "Dimensions of the popcorn graph". Proceedings of the American Mathematical Society. 150 (11): 4729–4742. arXiv:2007.08407. doi:10.1090/proc/15729.
7. Trifonov, Vladimir; Pasqualucci, Laura; Dalla-Favera, Riccardo; Rabadan, Raul (2011). "Fractal-like Distributions over the Rational Numbers in High-throughput Biological and Clinical Data". Scientific Reports. 1 (191): 191. arXiv:1010.4328. Bibcode:2011NatSR...1E.191T. doi:10.1038/srep00191. PMC 3240948. PMID 22355706.

References

• Thomae, J. (1875), Einleitung in die Theorie der bestimmten Integrale (in German), Halle a/S: Verlag von Louis Nebert
• Abbott, Stephen (2016), Understanding Analysis (Softcover reprint of the original 2nd ed.), New York: Springer, ISBN 978-1-4939-5026-3
• Bartle, Robert G.; Sherbert, Donald R. (1999), Introduction to Real Analysis (3rd ed.), Wiley, ISBN 978-0-471-32148-4 (Example 5.1.6 (h))
• Beanland, Kevin; Roberts, James W.; Stevenson, Craig (2009), "Modifications of Thomae's Function and Differentiability", The American Mathematical Monthly, 116 (6): 531–535, doi:10.4169/193009709x470425, JSTOR 40391145
• Dunham, William (2008), The Calculus Gallery: Masterpieces from Newton to Lebesgue (Paperback ed.), Princeton: Princeton University Press, ISBN 978-0-691-13626-4
• Spivak, M. (1965), Calculus on manifolds, Perseus Books, ISBN 978-0-8053-9021-6

External links

• "Dirichlet-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Dirichlet Function". MathWorld.