On Baire one Darboux functions with Lusin's condition (N) (Q788132)

In this paper is given the following monotonicity theorem: If f is a real function defined on an interval I and if f satisfies the following properties: (i) f is a Darboux, Baire one function, (ii) f fulfills Lusin's condition (N) on I, (iii)f'(x)\(\geq 0\) holds at almost all points at which f has a finite derivative, then f is a continuous non-decreasing function on I. - This theorem generalizes a theorem mentioned in \textit{S. Saks's} book ''Theory of the integral'' (1937; Zbl 0017.30004) (p. 286) in which the condition (i) is replaced by the continuity of f. The proof of the author's theorem is based on a theorem of Bruckner [Theorem 2.2 in the book by \textit{A. M. Bruckner}, Differentiation of real functions (1978; Zbl 0382.26002), p. 178] and on the following theorem of Ellis: Each measurable function on an interval I fulfilling Lusin's condition (N) on I fulfills Banach's condition \((T_ 2)\) [\textit{H. W. Ellis}, Can. J. Math. 3, 471-484 (1951; Zbl 0044.055)]. The detailed proof of the theorem, as some examples which complete its meaning, as some of its applications will appear in the author's paper: ''On Bernoulli-L'Hospital- Ostrowski rules and monotonicity theorems''.