24 | 07 | 2017


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Article 18

A fundamental lemma for monotonicity

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

Mathematical reviews subject classification: 26A48, 26A24.



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 Cd.

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 Mémoire sur la représentation finie des fonctions continues, Math. Ann., 103 (1930), p. 286), where (N) is replaced by (M) and the continuity by Ci

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.