Equivalent Definition of a support of a function $$Supp(f(x))= \overline{\{x \in \mathbb{R} |f(x) \neq 0\}}$$
My question:
How is the support of the function f, equals $ \{f^\mathrm{o}=0\}^c$? where $f^\mathrm{o}$ is the interior of $f$.
My Approach
$$Supp(f(x))= \overline{\{x \in \mathbb{R} |f(x) \neq 0\}} = \overline{\{x \in \mathbb{R} |f(x)\gt 0 \text{ or } f(x) \lt 0\}}$$
If we take the compliment we get, $$\overline{\{x \in \mathbb{R} |f(x)\gt 0 \text{ or } f(x) \lt 0\}}^c= \big[\big(\{x \in \mathbb{R} |f(x)\gt 0 \text{ or } f(x) \lt 0\}\big)^c\big]^0$$.
How to proceed from this step to get the final result?

In any topological space we always have $(A^{0})^{c}=\overline {A^{c}}$. Take $A=\{x:f(x)=0\}$. 
