In my real analysis homework, theres a question which begins with
"let $F$ be a distribution on $\mathbb{R}^n$ such that $\operatorname{supp}(F)=\{0\}$"
This means that the only $\phi \in C_c^\infty(\mathbb{R}^n)$ such that $\langle F,\phi\rangle\neq 0$ are the $C_c^\infty(\mathbb{R}^n)$ functions whose support is $0$, meaning they are zero everywhere except perhaps at $0$? In particular, $F$'s only nonzero values come from the delta function or zero functions?
I'm confused and any clarification would be much appreciated.
By definition, $F$ has support $\{0\}$ if $\langle F,\phi\rangle=0$ for all $\phi$ with support not intersecting $0$. In other words, only functions $\phi$ with support intersecting $0$ may have $\langle F,\phi\rangle\ne0$.