Problem taken from the link: http://web.mit.edu/rwbarton/Public/func-eq.pdf I am stating the question here
For which $\alpha$ does there exists a nonconstant function $f: \mathbb{R} \to \mathbb{R}$ such that
\begin{equation*} f(\alpha(x+y))=f(x)+f(y)~\forall x,y \in \mathbb{R}. \end{equation*}
Clearly for $\alpha=1$ we see that this case is satisfied, by taking the identity function. But are there other values of $\alpha$ for which this condition is satisfied.