Thomae's function: Difference between revisions

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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 [[Fσ set|{{math|''F''_σ}} set]]. If such a function existed, then the irrationals would be an {{math|''F''_σ}} set. The irrationals would then be the [[countable set|countable]] [[union (set theory)|union]] of [[closed set]]s <math display="inline">\bigcup_{i = 0}^\infty C_i</math>, but since the irrationals do not contain an interval, neither can any of the <math>C_i</math>. Therefore, each of the <math>C_i</math> 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.

Thomae’s function is also closely related to random processes generated by events within discrete uniform distributions such as pi digits, live roulette spins and dice rolls. A random variable C{{sub|i}} is repeatedly sampled N times from a discrete uniform distribution, where i ranges from 1 to N. For instance, consider integer values ranging from 1 to 10 or 1 to 6 in the case of a fair dice. Moments of occurrence, T{{sub|k}}, signify when events C{{sub|i}} repeat, defined as C{{sub|i}} = C{{sub|i-1}} or C{{sub|i}} = C{{sub|i-2}}, where k ranges from 1 to M, with M being less than N. Subsequently, define S{{sub|j}} as the interval between successive T{{sub|k}}, representing the waiting time for an event to occur. The random variable S is geometrically distributed. Finally, introduce Z{{sub|l}} as ln(S{{sub|j}}) – ln(S{{sub|j-1}}), where l ranges from 1 to U-1. The random variable Z displays fractal properties, resembling the shape distribution akin to Thomae's or Dirichlet function.<ref>Endrit Dheskali.

Generation of a recurrent fractal process using discrete uniform distributions, 26 February 2024, PREPRINT (Version 1) available at Research Square [https://doi.org/10.21203/rs.3.rs-3987858/v1]</ref>

[[File:Thomae's function like distribution.jpg|Probability density function of the random variable Z]]

A variant of Thomae's function can be used to show that any {{math|''F''_σ}} subset of the real numbers can be the set of discontinuities of a function. If <math display="inline"> A = \bigcup_{n=1}^{\infty} F_n</math> is a countable union of closed sets <math> F_n</math>, define