In Advanced Calculus by Loomis and Sternberg, regarding functions $f : \mathbb{R}^{n} \rightarrow \mathbb{R}$, they frequently employ the condition that $f$“has compact support”, where the support of $f$ is defined as
by supp $f$, we shall mean the closure of the set where $f$ is not zero
After thinking a bit about the implications of this “$f$ has compact support” criterion, it occurred to me that this looks like just an overly complicated way of saying that $f$ is zero outside some bounded subset of $\mathbb{R}^{n}$
Is this right, or am I missing something? For functions $f : \mathbb{R}^{n} \rightarrow \mathbb{R}$, is “$f$ has compact support” in fact equivalent to “$\{\mathbf{x}: f(\mathbf{x}) \neq 0 \}$ is bounded”?