Real Analysis Exchange

On Infinite Unilateral Derivatives

F. S. Cater

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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

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

First available in Project Euclid: 18 December 2008

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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

unilateral derivative bilateral derivative metric space


Cater, F. S. On Infinite Unilateral Derivatives. Real Anal. Exchange 33 (2007), no. 2, 309--316.

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