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|f(x+y)-f(x)-f(y)|<=1, approximate by an additive function

Source: Turkey TST 2000

February 28, 2005

functioninductionalgebrainequalitiesTSTadditive functionfunctional equation

Problem Statement

Given is a function

f:\mathbb{R}\rightarrow \mathbb{R}

such that

|f(x+y)-f(x)-f(y)|\leq 1

. Prove the existence of an additive function

g:\mathbb{R}\rightarrow \mathbb{R}

(that is

g(x+y)=g(x)+g(y)

) such that

|f(x)-g(x)|\leq 1

for any

x \in \mathbb{R}