On Algebraic Properties of the Family of Weakly Świa̧tkowski Functions

Abstract

In the paper we will focus on weakly Światkowski functions. We say that f satisfies the weak Świa̧tkowski condition (or f is a weakly Świa̧tkowski function) if for all $$x_1, x_2$$ x 1 , x 2 with $$f(x_1)< f(x_2)$$ f ( x 1 ) < f ( x 2 ) there is a point x between $$x_1$$ x 1 and $$x_2$$ x 2 such that $$f(x_1)<f(x)< f(x_2)$$ f ( x 1 ) < f ( x ) < f ( x 2 ) . This definition is a modification of the Świa̧tkowski condition, in which point x mentioned above has to be a point of continuity of f. In the paper we will examine some properties of weakly Świa̧tkowski functions and consider lineability and algebrability of some families of functions related to them.