Thomae's function: Difference between revisions

Revision as of 19:35, 17 October 2022

Revision as of 10:03, 19 October 2022

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::The [[Lebesgue integrability condition|Lebesgue criterion for integrability]] states that a bounded function is Riemann integrable if and only if the set of all discontinuities has [[Lebesgue measure|measure zero]].<ref>{{Harvnb|Spivak|1965|p=53|loc=Theorem 3-8}}</ref> Every [[countability|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 <math>0</math> over any set because the function is equal to zero ''[[almost everywhere]]''.

*If <math>G = \{ \, (x,f(x)) : x \in (0,1) \, \} \subset \mathbb{R}^2</math> is the graph of the restriction of <math>f</math> to <math>(0,1)</math>, then the [[Minkowski–Bouligand dimension|'''box-counting dimension''']] of <math>G</math> is <math>4/3</math>.<ref>{{cite journal |last1=Chen |first1=Haipeng |last2=Fraser |first2=Jonathan M. |last3=Yu |first3=Han |year=2022 |title=Dimensions of the popcorn graph |journal=[[Proceedings of the American Mathematical Society]] |volume=150 |number=11 |pages=4729–4742 |doi=10.1090/proc/15729 |arxiv=2007.08407}}</ref>

==Related probability distributions==