# A fundamental lemma for monotonicity

### Real Analysis Exchange 19 (1993/4), 579-589.

#### Mathematical reviews subject classification: 26A48, 26A24.

## Abstract

The main result of this paper is Lemma 4, which is a generalization of Lemma 6 of Article 7.

The following theorem of Banach is well known (see S. Saks, *Theory of the integral*, 2nd. rev. ed., vol. PWN, Monografie Matematyczne, Warsaw, 1937, p. 286):

Any function which is continuous and satisfies Lusin's condition (N) on an interval, is derivable at every point of a set of positive measure.

Foran generalizes this result (see *A generalization of absolute continuity, *Real Analysis Exchange **5** (1979/80), 82-91), showing that Banch's theorem remains true if condition (N) is replaced by Foran's condition (M).

An improvement of Foran's theorem is given in Article 7 (condition (M) is replaced by condition M).

In our article, using Lemma 4, we improve this last result in Theorem 5, replacing the continuity by condition C_{d}.

Theorem 5 is then used to prove a monotonicity theorem which permits an extension (see Corollary 1) of Corollary 3 of Article 7.

Both, the corollary of Article 7 and Corollary 1 extend the following theorem of Nina Bary (see *MÃ©moire sur la reprÃ©sentation finie des fonctions continues*, Math. Ann., **103 **(1930), p. 286), where (N) is replaced by (M) and the continuity by C_{i}:

Every continuous function which satisfies Lusin's condition (N) and whose derivative is nonnegative at almost every point x where F is derivable, is monotone nondecreasing.