I am reading Rudin's PMA and I have a question about the definition of the support.
Definition The support of a function $f$ on $ \Bbb R^{k}$ is the closure of the set of all points $ x \in \Bbb R^{k}$ at which $f(x) \neq 0$
My question is that, do we require $f(x) \neq 0$ on the support? For example, suppose we have a function $f$ defined in $\Bbb R^{k}$, $f(x) \neq 0$ in an open ball $N_{r}(0)$. By the definition, the support of the function is the union of the open ball and its surface. Do we also require $f(x)\neq0$ at points on the surface?
Thanks in advance
No, it is not required that $f(x)\neq 0$ on the whole support. If it was so, then the support would be both open and closed, and therefore would be a connected component of the domain of $f$, which is only a very particular case whereas the notion of support of a function is useful in many other cases.
The fact that the support is required to be closed allows you to talk about functions with compact support, for instance, which are useful in differential geometry.