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Let X be a real linear topological space. We characterize solutions f : X → R and M : R → R of the equation f(x + M(f(x))y) = f(x)f(y) under the assumption that f and M have the Darboux property. Let X be a real linear topological space. We know that a function f : X → R having the Darboux property and satisfying the exponential equation (1) f(x + y) = f(x)f(y) need not be continuous (see [9], cf. also [4, Remark 2]), and also that every solution f : X → R of the Gołąb–Schinzel equation (2) f(x + f(x)y) = f(x)f(y) having the Darboux property is continuous (see [4]). These two classical functional equations, in spite of their different nature, can be “connected” in the following equation: (3) f(x + M(f(x))y) = f(x)f(y), i.e. (1) and (2) are particular cases of (3) with M = 1 and M = idR, respectively. So, the study of the solutions f : X → R, M : R → R of (3) having the Darboux property seems to be interesting. Equation (1) is well known; for further information on it see e.g. [2]. The first paper concerning (2) is due to S. Gołąb and A. Schinzel [7]. This equation and its generalizations have been studied by many authors (the relevant bibliography can be found in [6]). In particular, J. Brzdęk has studied extensively the equation (4) f(x + f(x)y) = f(x)f(y) for a positive integer n, which is tightly connected with some classes of subgroups of the Lie groups Ls (see [3]) and some other groups. Referring to the second part of Hilbert’s 2000 Mathematics Subject Classification: 39B52, 26A30.

Functions having the Darboux property and satisfying some functional equation