Does Global Lipschitz Continuous imply boundedness? 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. 

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.
