Open Access
2004-2005 On continuous N-functions and an example of Mazurkiewicz
F. S. Cater
Real Anal. Exchange 30(1): 201-206 (2004-2005).


Let $f$ and $g$ be continuous real functions on the interval $[a,b]$, and let $K$ denote the set of all knot points of $f$. Let $E$ be a set of measure zero for which $f(E)$ has measure zero and $(f+g)(E)$ does not, and let $g$ be differentiable at each point of $E$ closure. We prove that $K$ must meet $E$, and moreover the intersection of $K$ with the closure of $E$ must contain a nonvoid perfect subset. Thus in particular, the function of Mazurkiewicz is a continuous N-Function with as many knot points as there are real numbers.


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F. S. Cater. "On continuous N-functions and an example of Mazurkiewicz." Real Anal. Exchange 30 (1) 201 - 206, 2004-2005.


Published: 2004-2005
First available in Project Euclid: 27 July 2005

zbMATH: 1060.26007
MathSciNet: MR2127526

Primary: 26A24 , 26A27 , 26A45 , 26A46

Keywords: absolutely continuous , Bounded variation , continuous N-function , Dini derivate , knot point , measure

Rights: Copyright © 2004 Michigan State University Press

Vol.30 • No. 1 • 2004-2005
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