I am looking for a function $f:\mathbb{R}\to\mathbb{R}$ such that it admits $$f(x+y)=f(x)+f(y)$$ for any $x,y\in\mathbb{R}$ but there exist $\alpha,x\in\mathbb{R}$ such that $$f(\alpha x)\neq\alpha f(x).$$ All I know is that the function is hard to contruct.
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3See this. – David Mitra Sep 13 '17 at 06:27
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What happen if we consider $\mathbb{R}$ as a vector space over $\mathbb{R}$? Can you give a short comment? – Sukan Sep 13 '17 at 06:34
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@Sukan nothing happens, it is a one-dimensional vector space, hence boring ;) – daw Sep 13 '17 at 14:55