A basic question about $\operatorname{supp}f$ (support of f).

Question

Is it true that $\operatorname{supp}f$ is the complement of the biggest open set where $f=0 $?

Here $\operatorname{supp}f=$ {$x\in \Bbb R^n ; f(x)\not=0$} and $f\in C$ (collection of continuous maps from $\Bbb R^n \to \Bbb R$)

Answer 1

Since your question is tagged $L^p$-spaces (which are spaces of equivalence classes of functions), one should distinguish between

• Support of a function $f$, which is the complement of the largest open set on which $f$ is zero identically.
• Support of an equivalence class $f\in L^p$, which is the complement of the largest open set on which (every representative of) $f$ is zero almost everywhere.

In first case, the existence of largest such open set follows by taking the union of all open sets that qualify. In the second case one has to be more careful: assume that there is a countable base of the topology and take the union of all base elements that meet the requirement.

The second notion of support is sometimes called essential support, I think.