Real Analysis Exchange

On the Derivatives of Functions of Bounded Variation

F. S. Cater

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Abstract

Using a standard complete metric $w$ on the set $F$ of continuous functions of bounded variation on the interval $[0,1]$, we find that a typical function in $F$ has an infinite derivative at continuum many points in every subinterval of $[0,1]$. Moreover, for a typical function in $F$, there are continuum many points in every subinterval of $[0,1]$ where it has no derivative, finite nor infinite. The restriction of the derivative of a typical function in $F$ to the set of points of differentiability has infinite oscillation at each point of this set.

Article information

Source
Real Anal. Exchange, Volume 26, Number 2 (2000), 923-932.

Dates
First available in Project Euclid: 27 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1214571382

Mathematical Reviews number (MathSciNet)
MR1844408

Zentralblatt MATH identifier
1012.26005

Subjects
Primary: 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05] 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives 26A30: Singular functions, Cantor functions, functions with other special properties 26A45: Functions of bounded variation, generalizations 26A46: Absolutely continuous functions 26A48: Monotonic functions, generalizations

Keywords
bounded variation absolutely continuous singular derivative complete metric category

Citation

Cater, F. S. On the Derivatives of Functions of Bounded Variation. Real Anal. Exchange 26 (2000), no. 2, 923--932. https://projecteuclid.org/euclid.rae/1214571382


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References

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