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Interesting Cauchy function with period on a one-variable restriction

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December 13, 2019
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Problem Statement

Prove that there exists a nonconstant function f:R2⟶R f:\mathbb{R}^2\longrightarrow\mathbb{R} verifying the following system of relations: {f(x,x+y)=f(x,y),emsp;∀x,y∈Rf(x,y+z)=f(x,y)+f(x,z),emsp;∀x,y∈R \left\{ \begin{matrix} f(x,x+y)=f(x,y) ,&   \forall x,y\in\mathbb{R} \\f(x,y+z)=f(x,y) +f(x,z) ,&   \forall x,y\in\mathbb{R} \end{matrix} \right.
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