# Monotonicity theorems

### Real Analysis Exchange 12 (1987), 420-454.

#### Mathematical reviews subject classification: 26A48

## Abstract

In the article *An affirmative answer to a problem of Zahorski **and some consequences* (Mich. Math. J., **13** (1966), 15-26), Bruckner proved the following theorem:

Let f be a function, satisfying the following conditions on an interval [a,b]:

(i) f is a Darboux function, in Baire's class one;

(ii) f is VBG;

(iii) f is increasing on each closed subinterval of [a,b] on which it is continuous an VB.

Then f is continuous and nondecreasing on [a,b].

In this paper we generalize Bruckner's theorem (our proof being shorter). Then we give applications of this theorem, which generalize some consequences of Bruckner's theorem.

The following theorem of Banach is well known:

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

Of course the condition (N) implies condition T_{2}, and it is this fact that leads to the proof of Banach's theorem.

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

An improvement of Foran's theorem is given in Theorem 9, which is then used to prove a monotonicity theorem that generalize a result of Nina Bary (see *Memoire sur la representation finie des **fonctions continues*, Math. Ann., **103** (1930), p. 200).

Further we give many applications of Theorem 10.