0 Irrational 1 Rational Continuous Nowhere

Function that is discontinuous at rationals and continuous at irrationals

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

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

f ( x ) = { 1 q if x = p q ( x  is rational), with p Z  and q N  coprime 0 if x  is irrational. {\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) p Z {\displaystyle p\in \mathbb {Z} } and q N {\displaystyle q\in \mathbb {N} } , the function is well-defined. Note that q = + 1 {\displaystyle q=+1} is the only number in N {\displaystyle \mathbb {N} } that is coprime to p = 0. {\displaystyle p=0.}

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

Properties [edit]

Proof of periodicity

For all x R Q , {\displaystyle x\in \mathbb {R} \smallsetminus \mathbb {Q} ,} we also have x + n R Q {\displaystyle x+n\in \mathbb {R} \smallsetminus \mathbb {Q} } and hence f ( x + n ) = f ( x ) = 0 , {\displaystyle f(x+n)=f(x)=0,}

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

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

Proof of discontinuity at rational numbers

Let x 0 = p / q {\displaystyle x_{0}=p/q} be an arbitrary rational number, with p Z , q N , {\displaystyle \;p\in \mathbb {Z} ,\;q\in \mathbb {N} ,} and p {\displaystyle p} and q {\displaystyle q} coprime.

This establishes f ( x 0 ) = 1 / q . {\displaystyle f(x_{0})=1/q.}

Let α R Q {\displaystyle \;\alpha \in \mathbb {R} \smallsetminus \mathbb {Q} \;} be any irrational number and define x n = x 0 + α n {\displaystyle x_{n}=x_{0}+{\frac {\alpha }{n}}} for all n N . {\displaystyle n\in \mathbb {N} .}

These x n {\displaystyle x_{n}} are all irrational, and so f ( x n ) = 0 {\displaystyle f(x_{n})=0} for all n N . {\displaystyle n\in \mathbb {N} .}

This implies | x 0 x n | = α n , {\displaystyle |x_{0}-x_{n}|={\frac {\alpha }{n}},\quad } and | f ( x 0 ) f ( x n ) | = 1 q . {\displaystyle \quad |f(x_{0})-f(x_{n})|={\frac {1}{q}}.}

Let ε = 1 / q {\displaystyle \;\varepsilon =1/q\;} , and given δ > 0 {\displaystyle \delta >0} let n = 1 + α δ . {\displaystyle n=1+\left\lceil {\frac {\alpha }{\delta }}\right\rceil .} For the corresponding x n {\displaystyle \;x_{n}} we have

| f ( x 0 ) f ( x n ) | = 1 / q ε {\displaystyle |f(x_{0})-f(x_{n})|=1/q\geq \varepsilon \quad } and

| x 0 x n | = α n = α 1 + α δ < α α δ δ , {\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 f {\displaystyle f} at x 0 {\displaystyle x_{0}} .

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

Proof of continuity at irrational arguments

Since f {\displaystyle f} is periodic with period 1 {\displaystyle 1} and 0 Q , {\displaystyle 0\in \mathbb {Q} ,} it suffices to check all irrational points in I = ( 0 , 1 ) . {\displaystyle I=(0,\;1).\;} Assume now ε > 0 , i N {\displaystyle \varepsilon >0,\;i\in \mathbb {N} } and x 0 I Q . {\displaystyle x_{0}\in I\smallsetminus \mathbb {Q} .} According to the Archimedean property of the reals, there exists r N {\displaystyle r\in \mathbb {N} } with 1 / r < ε , {\displaystyle 1/r<\varepsilon ,} and there exist k i N , {\displaystyle \;k_{i}\in \mathbb {N} ,} such that

for i = 1 , , r {\displaystyle i=1,\ldots ,r} we have 0 < k i i < x 0 < k i + 1 i . {\displaystyle 0<{\frac {k_{i}}{i}}<x_{0}<{\frac {k_{i}+1}{i}}.}

The minimal distance of x 0 {\displaystyle x_{0}} to its i-th lower and upper bounds equals

d i := min { | x 0 k i i | , | x 0 k i + 1 i | } . {\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 δ {\displaystyle \delta } as the minimum of all the finitely many d i . {\displaystyle d_{i}.}

δ := min 1 i r { d i } , {\displaystyle \delta :=\min _{1\leq i\leq r}\{d_{i}\},\;} so that

for all i = 1 , . . . , r , {\displaystyle i=1,...,r,} | x 0 k i / i | δ {\displaystyle \quad |x_{0}-k_{i}/i|\geq \delta \quad } and | x 0 ( k i + 1 ) / i | δ . {\displaystyle \quad |x_{0}-(k_{i}+1)/i|\geq \delta .}

This is to say, all these rational numbers k i / i , ( k i + 1 ) / i , {\displaystyle k_{i}/i,\;(k_{i}+1)/i,\;} are outside the δ {\displaystyle \delta } -neighborhood of x 0 . {\displaystyle x_{0}.}

Now let x Q ( x 0 δ , x 0 + δ ) {\displaystyle x\in \mathbb {Q} \cap (x_{0}-\delta ,x_{0}+\delta )} with the unique representation x = p / q {\displaystyle x=p/q} where p , q N {\displaystyle p,q\in \mathbb {N} } are coprime. Then, necessarily, q > r , {\displaystyle q>r,\;} and therefore,

f ( x ) = 1 / q < 1 / r < ε . {\displaystyle f(x)=1/q<1/r<\varepsilon .}

Likewise, for all irrational x I , f ( x ) = 0 = f ( x 0 ) , {\displaystyle x\in I,\;f(x)=0=f(x_{0}),\;} and thus, if ε > 0 {\displaystyle \varepsilon >0} then any choice of (sufficiently small) δ > 0 {\displaystyle \delta >0} gives

| x x 0 | < δ | f ( x 0 ) f ( x ) | = f ( x ) < ε . {\displaystyle |x-x_{0}|<\delta \implies |f(x_{0})-f(x)|=f(x)<\varepsilon .}

Therefore, f {\displaystyle f} is continuous on R Q . {\displaystyle \mathbb {R} \smallsetminus \mathbb {Q} .\quad }

  • f {\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 ( a n ) n = 1 {\displaystyle (a_{n})_{n=1}^{\infty }} with a n x 0 {\displaystyle a_{n}\neq x_{0}} for all n N + {\displaystyle n\in \mathbb {N} _{+}} that converges to the irrational point x 0 , {\displaystyle x_{0},\;} the sequence ( f ( a n ) ) n = 1 {\displaystyle (f(a_{n}))_{n=1}^{\infty }} is identically 0 , {\displaystyle 0,\;} and so lim n | f ( a n ) f ( x 0 ) a n x 0 | = 0. {\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 ( b n ) n = 1 = ( k n / n ) n = 1 , {\displaystyle (b_{n})_{n=1}^{\infty }=(k_{n}/n)_{n=1}^{\infty },\;} converging to x 0 , {\displaystyle x_{0},\;} with k n Z {\displaystyle k_{n}\in \mathbb {Z} } and n N {\displaystyle n\in \mathbb {N} } coprime and | k n / n x 0 | < 1 5 n 2 . {\displaystyle |k_{n}/n-x_{0}|<{\frac {1}{{\sqrt {5}}\cdot n^{2}}}.\;}
Thus for all n , {\displaystyle n,} | f ( b n ) f ( x 0 ) b n x 0 | > 1 / n 0 1 / ( 5 n 2 ) = 5 n 0 {\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 f {\displaystyle f} is not differentiable at all irrational x 0 . {\displaystyle x_{0}.}
  • f {\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 f {\displaystyle f} has maxima.
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 0 {\displaystyle 0} over any set because the function is equal to zero almost everywhere.

[edit]

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 m , n {\displaystyle m,n} are sampled from a distribution f ( n , m ) {\displaystyle f(n,m)} and used to generate ratios q = n / ( n + m ) {\displaystyle q=n/(n+m)} , this gives rise to a distribution g ( q ) {\displaystyle 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 ) ) = t = 1 f ( t a ) f ( t b ) {\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 f ( k ) = k α e β k / L i α ( e β ) {\displaystyle f(k)=k^{-\alpha }e^{-\beta k}/\mathrm {Li} _{\alpha }(e^{-\beta })} (where L i α {\displaystyle \mathrm {Li} _{\alpha }} is the polylogarithm function) then g ( a / ( a + b ) ) = ( a b ) α L i 2 α ( e ( a + b ) β ) / L i α 2 ( e β ) {\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 { 1 , 2 , , L } {\displaystyle \{1,2,\ldots ,L\}} g ( a / ( a + b ) ) = ( 1 / L 2 ) L / max ( a , b ) {\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 [edit]

For integers, the exponent of the highest power of 2 dividing n {\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.

[edit]

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 i = 0 C i {\displaystyle \textstyle \bigcup _{i=0}^{\infty }C_{i}} , but since the irrationals do not contain an interval, neither can any of the C i {\displaystyle C_{i}} . Therefore, each of the C i {\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 A = n = 1 F n {\displaystyle A=\textstyle \bigcup _{n=1}^{\infty }F_{n}} is a countable union of closed sets F n {\displaystyle F_{n}} , define

f A ( x ) = { 1 n if x  is rational and n  is minimal so that x F n 1 n if x  is irrational and n  is minimal so that x F n 0 if x A {\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 f A {\displaystyle f_{A}} has A as its set of discontinuities.

See also [edit]

  • 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 [edit]

  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. ^ a b 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. PMC3240948. PMID 22355706.

References [edit]

  • 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, ISBN978-1-4939-5026-3
  • Bartle, Robert G.; Sherbert, Donald R. (1999), Introduction to Real Analysis (3rd ed.), Wiley, ISBN978-0-471-32148-4 (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, ISBN978-0-691-13626-4
  • Spivak, M. (1965), Calculus on manifolds, Perseus Books, ISBN978-0-8053-9021-6

External links [edit]

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

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Source: https://en.wikipedia.org/wiki/Thomae%27s_function

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