Fall 2015, problem 4

Let $f\colon \mathbb{R}^2 \to \mathbb{R}$ be a function such that for any square $S$ in $\mathbb{R}^2$ with vertices $v_1,v_2,v_3,v_4$, we have

$$ \sum_{j=1}^4 f(v_i) = 0. $$

Prove that $f$ is identically 0.

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