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f:R→R; f(x+y)=f(x)+f(y)+1, Show that there exists g:R→R

Source: Turkey TST 2000 P6

March 12, 2011

functionlimitinequalitiesalgebra

Problem Statement

Suppose

f:\mathbb{R} \to \mathbb{R}

is a function such that

|f(x+y)-f(x)-f(y)|\le 1\ \ \ \text{for all} \ \ x, y \in\mathbb R.

Prove that there is a function

g:\mathbb{R}\to\mathbb{R}

such that

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

and

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

for all

x,y \in\mathbb R.