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Les schémas de subdivision de Besicovitch et de Cantor

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Abstract

We introduce the Besicovitch and Cantor subdivision schemes with various examples. We identify two Besicovitch schemes whose limit fonctions are the Besicovitch function B and the Singh function S. It is recognized that both functions have at any point neither a right derivative, nor a left derivative, finite or infinite. We discover that B at point 4/7 has an infinite right derivative contrary to what is recognized. Nonetheless, we confirm that S has no one-sided derivatives, neither finite nor infinite.

Résumé

Nous définissons les schémas de subdivision de Besicovitch et de Cantor et nous les illustrons par divers exemples. Nous distinguons deux schémas de Besicovitch dont les fonctions limites sont respectivement une fonction B due à Besicovitch et une autre S due à Singh. Il est reconnu que ces deux fonctions n’admettent en aucun point une dérivée ni à gauche, ni à droite, finie ou infinie. Nous découvrons que B au point 4/7 a une dérivée à droite infinie contrairement à ce qui est reconnu. Néanmoins, nous confirmons que S n’a aucune dérivée unilatérale ni finie, ni infinie.

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  1. Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, Succ. Centre-ville, Montréal, QC, H3C 3J7, Canada

    Serge Dubuc

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  1. Serge Dubuc
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Correspondence to Serge Dubuc.

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Appendix Pseudo-code pour le tracé de la fonction de Singh

Appendix Pseudo-code pour le tracé de la fonction de Singh

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Dubuc, S. Les schémas de subdivision de Besicovitch et de Cantor. Ann. Math. Québec 44, 329–347 (2020). https://doi.org/10.1007/s40316-020-00131-9

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  • DOI: https://doi.org/10.1007/s40316-020-00131-9

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