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

Is it always true that \(g(2x)=2g(x)?\)

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

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