How can we prove multivariable polynomials are bounded on a closed set? the boundedness theorem is for single variable functions. Does an extension theorem exist?
Thank you.
2026-04-07 08:02:20.1775548940
Boundedness of multivariable polynomials
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They aren't in general! If the closed set in question is $\mathbb R^n$, then they are unbounded.
If the set is bounded, you can prove it in the same way for a single variable.
Assume $f$ is not bounded on your closed, bounded (i.e. compact) set $A$. Then for every $n \in \mathbb N$, there is $x_n \in A$ such that $f(x_n) > n$. Then $x_n$ is a bounded sequence, and so it must have a subsequential limit in $A$. Use continuity of $f$ to get a contradiction.