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EN
A topological space is called connected if it is not the union of two disjoint, nonempty and open sets in this space. The standard exercises show that here the concept of open sets can be replaced by closed sets or separated sets. In this context we will discuss the definition of connected sets in topological spaces, not being the whole space with particular regard to metric spaces, without the term of subspace topology.
EN
We consider the class of rational functions defined by the formula F(x, y) = φˉ¹(φ(x)φ(y)), where φ is a homographic function and we describe associative functions of the above form.
EN
We deal with a functional equation of the form ƒ(x + y) = F(ƒ(x), ƒ(y)) (so called addition formula) assuming that the given binary operation F is associative but its domain is not connected. The aim of the present paper is to discuss solutions of the equation [formula]. It turns out that this functional equation characterized an inverse proportionality type function, but if the domain of the unknown function has no neutral element. In this paper we admit fairly general structure in the domain of the unknown function.
EN
We consider the class of rational functions defined by the formula F(x, y) = ϕ −1 (ϕ(x) + ϕ(y)), where ϕ is a homographic function and we describe all associative functions of the above form.
EN
We deal with the functional equation (so called addition formula) of the form f(x + y) = F(f(x),f(y)), where F is an associative rational function. The class of associative rational functions was described by A. Chéritat [1] and his work was followed by a paper of the author. For function F deﬁned by F(x,y) = ϕ−1(ϕ(x) + ϕ(y)), where ϕ is a homographic function, the addition formula is fulﬁlled by homographic type functions. We consider the class of the associative rational functions defined by formula F(u,v) =uv αuv + u + v, where α is a ﬁxed real numer.
6
A characterization of a homographic type function II
EN
This article is a continuation of the investigations contained in the previous paper [2]. We deal with the following conditional functional equation: [wzór] implies [wzór] with λ ≠ 0.
7
A characterization of a homographic type function
EN
We deal with a functional equation of the form f(x + y) = F(f(x),f(y)) (the so called addition formula) assuming that the given binary operation F is associative but its domain of definition is not necessarily connected. In the present paper we shall restrict our consideration to the case when [formula]. These considerations may be viewed as counter parts of Losonczi's [7] and Domańska's [3] results on local solutions of the functional equation f(F(x, y)) = f(x) + f(y) with the same behaviour of the given associative operation F. In this paper we admit fairly general structure in the domain of the unknown function.
8
On Associative Rational Functions
EN
We deal with the following problem: which rational functions of two variables are associative? We shall determine all of them provided that at least of the coefficients in question vanishes.
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