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Another take on Cauchy's functional equation

Source: Miklós Schweitzer 2013, P7

July 12, 2014

functionreal analysisreal analysis unsolved

Problem Statement

Suppose that

{f: \Bbb{R} \rightarrow \Bbb{R}}

is an additive function

(

that is

{f(x+y) = f(x)+f(y)}

for all

{x, y \in \Bbb{R}})

for which

{x \mapsto f(x)f(\sqrt{1-x^2})}

is bounded of some nonempty subinterval of

{(0,1)}

. Prove that

{f}

is continuous.
Proposed by Zoltán Boros

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