Remarks on real-valued quasi-isometries

Abstract

In this paper we consider the class of solutions of the functional inequality | |f(x) - f( y)| - |x – y| | ≤ ε ϵ min{|f(x) - f(y)|,|x - y|}, x, y ϵ R where f : IR -> IR is the unknown function and ε ϵ [0,1). The two main tasks of the paper are to compare the above class of quasi-isometries with the class of quasi-additive functions and to answer the question of stability.