## Real Analysis Exchange

### On Infinite Unilateral Derivatives

F. S. Cater

#### 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

http://projecteuclid.org/euclid.rae/1229619409

Mathematical Reviews number (MathSciNet)
MR2458248

Zentralblatt MATH identifier
1158.26003

#### Citation

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

#### References

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• 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.