# Teoreme de monotonie pentru functii reale cu o variabila reala

### Studii si Cercetari Matematice 46 (1994), no. 3, 347-386, Ph. thesis (in Romanian).

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

## Abstract

This thesis is heavily based on Article 7, Article 8 and Article 4.

In *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]:

- f is a Darboux function, in Baire's class one;
- f is VBG;
- 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].

Bruckner obtained this result while answering affirmatively a problem presented by Zahorski in *Sur la première derivée* (Trans. Amer. Math. Soc., **69** (1950), 1-54).

This question was also answered independently by Swiatkowski in *On the conditions of monotonicity of functions* (Fund. Math., **59** (1966), 189-201).

In Chapter III we generalize Bruckner's theorem as follows:

Let f be a function satisfying the following conditions on [0,1]:

- f ∈ D ∩ (-);
- f ∈ B' on H = {x : f is continuous at x};
- f ∈ B
_{i} on U(f) = int(H).

Then f is continuous and increasing on [0,1].

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.

Foran extends this result (see *A generalization of absolute continuity,* Real Analysis Exchange **5** (1979/80), 82-91), asking his condition (M) instead of (N).

In Chapter IV, we improve Foran's result as follows:

Let f:[0,1] \to R be a continuous function satisfying condition [M] on [0,1]. Then f is derivable on a set of positive measure.

Moreover, if there exist 0 ≤ a < b ≤ 1 such that f(a) < f(b) then |f(P)| > 0, where P = {x : f'(x) ≥ 0}.

This result is then used to prove a monotonicity theorem (Theorem 10), which is a generalization of a theorem of Nina Bary (see *Mémoire sur la représentation finie des fonctions continues*, Math. Ann., **103 ** (1930), p. 200).

Further we give many applications of Theorem 10.

Using Thomson's notion of a *local system *satisfying some *intersection conditions*, we give other interesting monotonicity theorems.