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Let $f:\mathbb{R^n}\to \mathbb{R}$. If $G_f$ is the graph of $f$ and $G_f$ is closed, does it imply that $f$ is bounded?

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Before you ask: Closedness of the graph does not even imply continuity, though that would at least be an implication that one might expect. –  Hagen von Eitzen Mar 2 '13 at 14:53
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No. Take $$ f:(x_1,\ldots,x_n)\longmapsto x_1. $$

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