Show that if $f\in C^0(\mathbb R)$, then its support is identical with the support of the distribution $\langle f,\phi \rangle=\int f\phi \,dx$

Question

Show that if $f\in C^0(\mathbb R)$, then its support is identical with the support of the distribution $\langle f,\phi \rangle=\int f\phi \, dx$, where $\phi\in C^\infty_c(\mathbb R^n)$

And is this true when $f\in L^{loc}_1(\mathbb R^n)$?

Answer 1

Hint For simplicity I will denote by $u_f$ the distribution.

Show that $ \mbox{supp}(f) = \mbox{supp}(u_f)$ by double inclusion.

$\subseteq$: If $x \in \mbox{supp(f)}$ and $x \in U$ open, then there exists some $y \in U$ such that $f(y) \neq 0$.

Next, by continuity, there exists some open $y \in V \subset U$ such that $f$ is positive (or negative depending on the sign of $f(y)$) on $V$. Now you can construct some $\phi$ with $\mbox{supp}(\phi) \subset V \subset U$ such that $u_f(\phi) \neq 0$.

Deduce from here that $x \in \mbox{supp}(u_f)$.

$\supseteq$ If $x \in \mbox{supp}(u_f)$ then for every open set $x \in U$ you can find some $\phi$ with $\mbox{supp} (\phi) \subset U$ and $u_f(\phi) \neq 0$.

Deduce from here that there exists some $y \in U$ so that $f(y) \neq 0$.