Article 19

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]:

1. f is a Darboux function, in Baire's class one;
2. f is VBG;
3. 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.