  Suppose $X$ is a compact Hausdorff space. In terms of topolocical properties
of $X$, we find topological conditions on $X$ that are equivalent to each of
the following: 1. every additive local multiplication on $C\left( X\right) $ is
a multiplication, 2. every additive local multiplication on $C_{R}\left(
X\right) $ is a multiplication, and 3. every additive map on $C\left( X\right)
$ that is zero-preserving (i.e., $f\left( x\right) =0$ implies $\left(
Tf\right) \left( x\right) =0$) has the form $T\left( f\right) =T\left( 1\right)
\operatorname{Re}f+T\left( i\right) \operatorname{Im}f$.
