Real Analysis Exchange

On Infinite Unilateral Derivatives

F. S. Cater

Full-text: Open access

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 $(N)$.

Article information

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

Dates
First available in Project Euclid: 18 December 2008

Permanent link to this document
http://projecteuclid.org/euclid.rae/1229619409

Mathematical Reviews number (MathSciNet)
MR2458248

Zentralblatt MATH identifier
1158.26003

Subjects
Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A45: Functions of bounded variation, generalizations

Keywords
unilateral derivative bilateral derivative metric space

Citation

Cater, F. S. On Infinite Unilateral Derivatives. Real Anal. Exchange 33 (2007), no. 2, 309--316. http://projecteuclid.org/euclid.rae/1229619409.


Export citation

References

  • S. Saks, On the Functions of Besicovich in the Space of Continuous Functions, Fund. Math., 19 (1932), 211–219.
  • S. Saks, Theory of the Integral, 2nd. Rev. Ed., Dover, New York, 1964.
  • F. S. Cater, An Elementary Proof of a Theorem on Unilateral Derivatives, Canad. Math. Bull., 29(3) (1986), 341–343.