Supremum norm in $\mathbb{R}^n$

Question

Let $I$ be an interval of $\mathbb R$ and $f$ a function defined on $I$, we have $$\|f\|_\infty=\sup\{|f(x)|;\ x\in I\}$$ This is the version for real functions of one variable of "supremum norm". What is the equivalent for multivariable functions ? Should $I$ be a compact, a closet set, an open set ?

Answer 1

The natural generalization is to take a compact set, if you this to be defined for any continuoues functions. In fact, a subset $I$ of $\Bbb R^n$ is compact if and only if every continuous function from $I$ into $\Bbb R$ is bounded.