Stability problem for number-theoretically multiplicative functions

We deal with the stability question for multiplicative mappings in the sense of number theory. It turns out that the conditional stability assumption:

$\vert f(xy)-f(x)f(y)\vert\leq\varepsilon$   for relatively prime $x,y$

implies that $f$ lies near to some number-theoretically multiplicative function. The domain of $f$ can be general enough to admit, in special cases, the reduction of our result to the well known J. A. Baker - J. Lawrence - F. Zorzitto superstability theorem.