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If a function $h$ were Lipschitz Continuous on $\mathbb{R}^n$, would that imply the function is bounded? I would assume so since Lipschitz Continuous implies Uniform Continuous. Then again, uniform continuous implies boundedness when you map a bounded open interval to $\mathbb{R}^n$.

Thank you for your time and I appreciate any feedback.

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1 Answer 1

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Consider the function $f:\mathbb R\to\mathbb R$ given by $f(x)=x$.

$f$ is Lipschitz, but unbounded. However a Lipschitz function is bounded on a bounded domain.

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