Real Analysis Exchange
- Real Anal. Exchange
- Volume 33, Number 2 (2007), 309-316.
On Infinite Unilateral Derivatives
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 $(N)$.
Real Anal. Exchange Volume 33, Number 2 (2007), 309-316.
First available in Project Euclid: 18 December 2008
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A45: Functions of bounded variation, generalizations
Cater, F. S. On Infinite Unilateral Derivatives. Real Anal. Exchange 33 (2007), no. 2, 309--316.https://projecteuclid.org/euclid.rae/1229619409