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Euge
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Let ##D_r\subset \mathbb{R}^2## be the disk of radius ##r## centered at the origin. If ##f : \mathbb{R}^2 \to [0,\infty)## is uniformly continuous such that ##\sup_{0 < r< \infty} \iint_{D_r} f(x,y)\, dx\, dy < \infty##, show that ##f(x,y) \to 0## as ##x^2 + y^2 \to \infty##.