Let $X$ be a metric space, $f:X\to\mathbb R_{\geq 0}$ is continuous. Suppose there exists two open sets $X_1$ and $X_2$ such that $X = \overline{X_1\cup X_2}$ and $f|_{X_1}>0, f|_{X_2} = 0$. How to prove that $\operatorname{supp}f = \overline{X_1}$?
For sure I know how to prove that $\overline{X_1}\subseteq \operatorname{supp}f$ - but I have problems with the other direction.