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2007/2008 On Infinite Unilateral Derivatives
F. S. Cater
Real Anal. Exchange 33(2): 309-316 (2007/2008).

Abstract

We prove that for any continuous real valued function $f$ on $[a,b]$ there exists a continuous function $K$ such that $K\!-\!f$ has bounded variation and $(K\!-\!f)^\prime = 0$ almost everywhere on $[a,b]$ and such that in any subinterval of $[a,b]$, $K$ has right derivative $\infty$ at continuum many points, $K$ has left derivative $\infty$ at continuum many points, $K$ has right derivative $-\infty$ at continuum many points, and $K$ has left derivative $-\infty$ at continuum many points. Furthermore, functions $K$ with these properties are dense in $C[a,b]$. We can assume the infinite derivatives of $K$ are bilateral if $f$ is of bounded variation on $[a,b]$ or if $f$ satisfies Lusin's condition $(N)$.

Citation

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F. S. Cater. "On Infinite Unilateral Derivatives." Real Anal. Exchange 33 (2) 309 - 316, 2007/2008.

Information

Published: 2007/2008
First available in Project Euclid: 18 December 2008

zbMATH: 1158.26003
MathSciNet: MR2458248

Subjects:
Primary: 26A24 , 26A45

Keywords: bilateral derivative , metric space , unilateral derivative

Rights: Copyright © 2007 Michigan State University Press

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