Annales Univ. Sci. Budapest., Sect. Comp. 40 (2013) 81–93 REMARKS ON SOME SYSTEMS OF TWO SIMULTANEOUS FUNCTIONAL INEQUALITIES Anna Bahyrycz (Krak´ow, Poland) Janusz Brzd¸ ek (Krak´ow, Poland) Dedicated to Professor Zolt´an Dar´ oczy and Professor Imre K´atai on the occasion of their 75th birthdays Communicated by Antal J´arai (Received March 29, 2013; accepted July 05, 2013) Abstract. We study the real (measurable and continuous at a point) func- tions that satisfy, almost everywhere, some systems of two simultaneous functional inequalities. In particular, we obtain generalizations and exten- sions of some earlier results of D. Krassowska, J. Matkowski, P. Montel, and T. Popoviciu. 1. Introduction In what follows N, Z, Q, and R denote, as usual the sets of positive integers, integers, rationals and reals, respectively. Moreover N 0 := N ∪{0}. Let a, b ∈ R \{0}, ab -1 / ∈ Q, ab < 0. P. Montel [13] (see also [15] and [12, p. 228]) proved that a function f : R → R, that is continuous at a point and satisfies the system of functional inequalities (1.1) f (x + a) ≤ f (x), f (x + b) ≤ f (x) x ∈ R, must be constant. A similar result for measurable functions has been proved in [3], where a more abstract approach is assumed. Key words and phrases : Functional inequality, microperiodic function, Lebesgue measurabil- ity, Baire property, σ-ideal. 2010 Mathematics Subject Classification : 39B72.