## Real Analysis Exchange

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

### On the Derivatives of Functions of Bounded Variation

#### 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