Real Analysis Exchange
- Real Anal. Exchange
- Volume 26, Number 2 (2000), 923-932.
On the Derivatives of Functions of Bounded Variation
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.
Real Anal. Exchange, Volume 26, Number 2 (2000), 923-932.
First available in Project Euclid: 27 June 2008
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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