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Integrability question with a function on a box in $\mathbb{R}^2$ (bounty added)

Let $f:\mathbb{R} \to\mathbb{R} \ be \ bounded,$ $\phi: \mathbb{R}^2 \to\mathbb{R}^2 $ be defined as $\phi(x,y)=(x,y+f(x))$ Prove that if for every bounded box $B\subset \mathbb{R}^2, \phi(B)$ ...