Solution
= Solution
\b[No.] For example, let
$$
f(z)=\max(1-|z|,0),
\qquad
g(z)=\max(1-|z-3|,0).
$$
Both functions are continuous and nonzero, but their supports lie in disjoint closed unit discs. Hence $f(z)g(z)=0$ for every $z\in\mathbb C$.
= Solution
\b[No.] For example, let
$$
f(z)=\max(1-|z|,0),
\qquad
g(z)=\max(1-|z-3|,0).
$$
Both functions are continuous and nonzero, but their supports lie in disjoint closed unit discs. Hence $f(z)g(z)=0$ for every $z\in\mathbb C$.