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Article 8

A theorem which implies both Tolstoff's theorem and Zahorski's theorem on monotonicity

Rev. Roumaine Math. Pures Appl. XXX (1987), no. 10, 121-129.

Mathematical reviews subject classification: 26A48

 

Abstract

Goldowski and Tonelli have established in 1928 and 1930 the following theorem:

Let F:I ⊂ R → R satisfying the following conditions on I:

    1.  F is continuous;
    2.  F' exists (finite or infinite) on I, with the possible exception of a denumerable subset of I;
    3.  F' ≥ 0 a.e.

Then F is nondecreasing on I.

 

In 1939, Tolstoff obtained the following improvement of this theorem:

Let F:I ⊂ R → R satisfying the following conditions on I: 

    1.   F is approximately continuous;
    2.   F'_ap exists (finite or infinite) on I, with the possible exception of a denumerable subset of I;
    3.   F'ap ≥ 0 a.e.

Then F is continuous and nondecreasing on I.

 

Another generalization of the Goldowski Tonelli Theorem was obtained by Zahorski in 1950:

Let F:I ⊂ R → R satisfying the following conditions on I:

    1. F has the Darboux property;
    2. F' exists (finite or infinite) on I, with the possible exception of a denumerable subset of I;
    3. F' ≥ 0 a.e.

Then F is continuous and nondecreasing on I.

 

Zahorski asked wether his theorem can be generalized. An affirmative answer was obtained by Bruckner and Swiatkowski (1965-1966):


Let F:I ⊂ R → R satisfying the following conditions on I:

    1. F is Darboux Baire one;
    2. F'ap exists (finite or infinite) on I, with the possible exception of a denumerable subset of I;
    3. F'ap ≥ 0 a.e.

Then F is continuous and nondecreasing on I.

 

In the present paper we prove the following theorem which also generalizes both, Tolstoff's theorem and Zahorski's theorem, but our proof is short and very similar to that of Goldowski and Tonelli.

Let F:I ⊂ R → R satisfying the following conditions on I:

    1. F is [CG] and uCM;
    2. F'ap exists (finite or infinite) on I, with the possible exception of a denumerable subset of I;
    3. F'ap ≥ 0 a.e.

Then F is nondecreasing on I.