real analysis - Let $g:mathbb{R}tomathbb{R}$ be a measurable function such that $g(x+y) =g(x)+g(y).$ Then $g(x) = g(1)x$ .

Let $g:\mathbb{R}\to\mathbb{R}$ be a measurable function such that
$$g(x+y) =g(x)+g(y).$$

How to prove that $g(x) = cx$ for some $c\in \mathbb{R}?$

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The main thing to do here relies upon the fact that such function should be continuous and therefore by natural argument the answer will follow.

Using this
Additivity + Measurability $\implies$ Continuity

Therefore I found out that there is nothing missing in this question.