  In this paper the following implication is verified for certain basic
algebraic curves: if the additive real function $f$ approximately (i.e., with a
bounded error) satisfies the derivation rule along the graph of the algebraic
curve in consideration, then $f$ can be represented as the sum of a derivation
and a linear function. When, instead of the additivity of $f$, it is assumed
that, in addition, the Cauchy difference of $f$ is bounded, a stability theorem
is obtained for such characterizations of derivations.
