The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...

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find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y \in \mathbb{R}^n$

find all continuous functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfying \begin{equation*} f(x+y)+f(x-y)=2f(x)+2f(y)~\forall x,y \in \mathbb{R}^n. \end{equation*} My attempt: I manage to show ...