AbstractGiven an involution fon (0;1);we prove that the set C(f) := f>0 : fis an involutiong is a closed multiplicative subgroup of (0;1) andtherefore C(f) is f1g, (0;1) or  Z = f n : n2 Zg for some >0, 6= 1.Moreover, we provide examples of involutions possessing each one of theabove types as the set C(f) and prove that the unique involutions fsuchthat C(f) = (0;1) are f(x) = cx ;c>0. 1 Introduction. Given a metric space X, by an involution we mean a continuous self{mapf: X!Xsuch that (ff)(x) = xfor any x2Xand fis not the identityon X. Involutions on X= R, the set of real numbers, are usually called stronginvolutions and have been studied for a variety of reasons [11]. Among them,we can mention that the identity f(f(x)) = xis one of the oldest functional Key Words: involutions, closed subgroups, di erence and functional equationsMathematical Reviews subject classi cation: Primary: 39B22; Secondary: 26A18Received by the editors January 2, 2008Communicated by: Emma D’Aniello