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It is shown that, under some general algebraic conditions on xed real numbers a;b;; , every solution f : R ! R of the system of functional inequalities f(x +a) f(x) +; f(x +b) f(x) + that is continuous at some point must be a linear function (up to an additive constant). Analogous results for three other similar simulta- neous systems are presented. An application to a characterization of L p -norm is given. 1. Introduction. Every subadditive function f :R! R, that is, such that f(x +y) f(x) +f(y); x;y2R; where R stands for the set of reals, satises the simultaneous system of functional inequalities of additive type:

A pair of linear functional inequalities and a characterization of Lp-norm

A pair of linear functional inequalities and a characterization of L^p-norm