{"title":"关于递增连续奇异函数具有非零有限导数的点集","authors":"Marta Kossaczká, L. Zajícek","doi":"10.14321/realanalexch.47.2.1638769133","DOIUrl":null,"url":null,"abstract":"Sanchez, Viader, Paradis and Carrillo (2016) proved that there exists an increasing continuous singular function $f$ on $[0,1]$ such that the set $A_f$ of points where $f$ has a nonzero finite derivative has Hausdorff dimension 1 in each subinterval of $[0,1]$. We prove a stronger (and optimal) result showing that a set $A_f$ as above can contain any prescribed $F_{\\sigma}$ null subset of $[0,1]$.","PeriodicalId":44674,"journal":{"name":"Real Analysis Exchange","volume":" ","pages":""},"PeriodicalIF":0.1000,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Set of Points at which an Increasing Continuous Singular Function has a Nonzero Finite Derivative\",\"authors\":\"Marta Kossaczká, L. Zajícek\",\"doi\":\"10.14321/realanalexch.47.2.1638769133\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Sanchez, Viader, Paradis and Carrillo (2016) proved that there exists an increasing continuous singular function $f$ on $[0,1]$ such that the set $A_f$ of points where $f$ has a nonzero finite derivative has Hausdorff dimension 1 in each subinterval of $[0,1]$. We prove a stronger (and optimal) result showing that a set $A_f$ as above can contain any prescribed $F_{\\\\sigma}$ null subset of $[0,1]$.\",\"PeriodicalId\":44674,\"journal\":{\"name\":\"Real Analysis Exchange\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.1000,\"publicationDate\":\"2021-11-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Real Analysis Exchange\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.14321/realanalexch.47.2.1638769133\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Real Analysis Exchange","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.14321/realanalexch.47.2.1638769133","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the Set of Points at which an Increasing Continuous Singular Function has a Nonzero Finite Derivative
Sanchez, Viader, Paradis and Carrillo (2016) proved that there exists an increasing continuous singular function $f$ on $[0,1]$ such that the set $A_f$ of points where $f$ has a nonzero finite derivative has Hausdorff dimension 1 in each subinterval of $[0,1]$. We prove a stronger (and optimal) result showing that a set $A_f$ as above can contain any prescribed $F_{\sigma}$ null subset of $[0,1]$.
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