Is there $U\subset \Bbb R^2$ with Lebesgue measure $0$ such that
$$f(x+y)=f(x)+f(y)$$ for all $(x, y)\in U$ implies $f(x+y)=f(x)+f(y)$ for all $(x, y)\in\Bbb R^2$ ?
Is there $U\subset \Bbb R^2$ with Lebesgue measure $0$ such that
$$f(x+y)=f(x)+f(y)$$ for all $(x, y)\in U$ implies $f(x+y)=f(x)+f(y)$ for all $(x, y)\in\Bbb R^2$ ?