Let \(f,g : \mathbb{R} \longrightarrow \mathbb{R}\) such that \(f(2x)=2f(x)\) and \(g(x) = x+ f\big(g(x)\big).\)

If \(2g(x)\) is in the range of \(g\) for all real \(x,\) then is it always true that \(g(2x)=2g(x)?\)

**Note**: \(f\) and \(g\) are not necessarily continuous over \(\mathbb{R}.\)

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