On Infinite Unilateral Derivatives

F. S. Cater
Source: Real Anal. Exchange Volume 33, Number 2 (2007), 309-316.

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 $(N)$.

First Page:
Primary Subjects: 26A24, 26A45
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.rae/1229619409
Mathematical Reviews number (MathSciNet): MR2458248
Zentralblatt MATH identifier: 1158.26003

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Mathematical Reviews (MathSciNet): MR167578
F. S. Cater, An Elementary Proof of a Theorem on Unilateral Derivatives, Canad. Math. Bull., 29(3) (1986), 341–343.
Mathematical Reviews (MathSciNet): MR846714
Digital Object Identifier: doi:10.4153/CMB-1986-052-5