## Real Analysis Exchange

- Real Anal. Exchange
- Volume 28, Number 1 (2002), 175-190.

### On approximate derivatives and Krzyzewski-Foran lemma.

#### Abstract

Let $E \subseteq {\mathbb R}$ and $g:E \to {\mathbb R}$. We show that if $|g(E)| = 0$, then $\underline{g}_{\,ap}(x) \le 0 \le \overline{g}_{ap}(x)$ almost everywhere on $E$, which immediately implies a lemma of Krzyzewski \cite{10} and Foran \cite{7}. The function $g$ is said to satisfy the inverse Lusin condition $(N^{-1})$ on $E$ if $|g^{-1}(H)| = 0$ for every $H \subseteq g(E)$ with $|H| = 0$. We prove that if $g^\prime_{ap}(x)$ exists almost everywhere on $E$, then $g$ is an $N^{-1}$-function if and only if $g_{ap}^\prime(x) \neq 0$ almost everywhere on $E$. We also improve upon Foran's \cite{7} chain rule for approximate derivatives, and obtain necessary and sufficient conditions for its validity almost everywhere.

#### Article information

**Source**

Real Anal. Exchange, Volume 28, Number 1 (2002), 175-190.

**Dates**

First available in Project Euclid: 12 June 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1150118748

**Mathematical Reviews number (MathSciNet)**

MR1973978

**Zentralblatt MATH identifier**

1059.26004

**Subjects**

Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A48: Monotonic functions, generalizations 26A46: Absolutely continuous functions 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence

**Keywords**

approximate derivative chain rule inverse Lusin condition ($N^{-1}$) measurable boundaries monotone (functions) absolutely continuous.

#### Citation

Sarkhel, D. N. On approximate derivatives and Krzyzewski-Foran lemma. Real Anal. Exchange 28 (2002), no. 1, 175--190. https://projecteuclid.org/euclid.rae/1150118748