SCHWEITZER 2013 Problem 7

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