Thomae's function

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

Thomae's function, named after Carl Johannes Thomae, has many names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function, the Riemann function, or the Stars over Babylon (John Horton Conway's name). This real-valued function f(x) of the real variable x is defined as:

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

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

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

Properties

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

Proof of periodicity

For all $x\in \mathbb {R} \smallsetminus \mathbb {Q} \;:\quad f(x+n)=f(x)=0,$ since $(x+n)\in \mathbb {R} \smallsetminus \mathbb {Q} \;\Rightarrow x\in \mathbb {R} \smallsetminus \mathbb {Q} .$ For all $x\in \mathbb {Q} ,\;$ there exist $p\in \mathbb {Z} \;$ and $\;q\in \mathbb {N} ,$ such that $\;x=p/q,\;$ and $\gcd(p,\;q)=1.$ According to Bézout's identity, for some $a,\;b\in \mathbb {Z}$ holds $\gcd \left((p+nq),\;q\right)=ap+(an+b)q=\gcd(p,\;q)=1,$ and thus $f(x+n)=f({\tfrac {p+nq}{q}})={\tfrac {1}{q}}=f({\tfrac {p}{q}})=f(x).$

• $f$ is discontinuous at all rational numbers, dense within the real numbers.

Proof of discontinuity at rational numbers

Assume an arbitrary rational $x_{0}=p/q,\;$ with $\;p\in \mathbb {Z} ,\;q\in \mathbb {N} \;$ and $\;(p,\;q)\,$ coprime. This establishes $\quad f(x_{0})=1/q.$ Let $\;\alpha \in \mathbb {R} \smallsetminus \mathbb {Q} \;$ be any irrational number and define for all $\;i\in \mathbb {N} ,\;$ $\quad x_{i}=x_{0}+{\frac {\alpha }{i}}.$ These $x_{i}$ are all irrational, and so $\quad f(x_{i})=0.$ This implies $\quad |x_{0}-x_{i}|={\frac {\alpha }{i}},\quad$ and $\quad |f(x_{0})-f(x_{i})|={\frac {1}{q}}.$ Taking $\;\varepsilon =1/q\;$ and $\;i=1+\left\lceil {\frac {\alpha }{\delta }}\right\rceil \;$ selects an $\;x_{i},\;$ such that $|f(x_{0})-f(x_{i})|=1/q\geq \varepsilon \quad$ and for all $\delta >0\;:\;$$|x_{0}-x_{i}|={\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 $f$ at $x_{0}$. This shows that $f$ is discontinuous on $\;\mathbb {Q} .$

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

Proof of continuity at irrational arguments

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

• $f$ is nowhere differentiable.

Proof of being nowhere differentiable

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

• $f$ has a strict local maximum at each rational number.

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

• $f$ is Riemann integrable on any interval and the integral evaluates to $0$ 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. 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 $0$ over any set because the function is equal to zero almost everywhere.

Follow-up

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 $\textstyle \bigcup _{i=0}^{\infty }C_{i}$, but since the irrationals do not contain an interval, nor can any of the $C_{i}$. Therefore, each of the $C_{i}$ would be nowhere dense, and the irrationals would 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: 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 $A=\textstyle \bigcup _{n=1}^{\infty }F_{n}$ is a countable union of closed sets $F_{n}$, define

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

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

For a general construction on arbitrary metric space, see this article Kim, Sung Soo. "A Characterization of the Set of Points of Continuity of a Real Function." American Mathematical Monthly 106.3 (1999): 258-259.

Related probability distributions

Empirical probability distributions related to Thomae's function appear in DNA sequencing. 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 $m,n$ are sampled from a distribution $f(n,m)$ and used to generate ratios $q=n/(n+m)$, this gives rise to a distribution $g(q)$ on the rational numbers. If the integers are independent the distribution can be viewed as a convolution over the rational numbers, $g(a/(a+b))=\sum _{t=1}^{\infty }f(ta)f(tb)$. Closed form solutions exist for power-law distributions with a cut-off. If $f(k)=k^{-\alpha }e^{-\beta k}/\mathrm {Li} _{\alpha }(e^{-\beta })$ (where $\mathrm {Li} _{\alpha }$ is the polylogarithm function) then $g(a/(a+b))=(ab)^{-\alpha }\mathrm {Li} _{2\alpha }(e^{-(a+b)\beta })/\mathrm {Li} _{\alpha }^{2}(e^{-\beta })$. In the case of uniform distributions on the set $\{1,2,\ldots ,L\}$ $g(a/(a+b))=(1/L^{2})\lfloor L/\max(a,b)\rfloor$, which is very similar to Thomae's function. Both their graphs have fractal dimension 3/2.

The ruler function

For integers, the exponent of the highest power of 2 dividing $n$ 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 0's 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 those rational numbers whose denominators are powers of 2.

See also

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

References

• 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, 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.