  Let $C$ be an affine plane curve. We consider additive functions $f:
K\rightarrow K$ for which $f(x)f(y)=0$, whenever $(x,y)\in C$. We show that if
$K=\mathbb{R}$ and $C$ is the hyperbola with defining equation $xy=1$, then
there exist nonzero additive functions with this property. Moreover, we show
that such a nonzero $f$ exists for a field $K$ if and only if $K$ is
transcendental over $\mathbb{Q}$ or over $\mathbb{F}_p$, the finite field with
$p$ elements. We also consider the general question when $K$ is a finite field.
We show that if the degree of the curve $C$ is large enough compared to the
characteristic of $K$, then $f$ must be identically zero.
