On a theorem of Baker, Lawrence and Zorzitto

The result of J. Baker, J. Lawrence and F. Zorzitto on the stability of the equation $f(x + y) = f(x)f(y)$ is generalized by proving the following theorem: if $G$ is a semigroup and $V$ is a right invariant linear space of complex valued functions on $G$, and if $f$, $m$ are complex valued functions on $G$ for which the function $x \to f(xy) - f(x)m(y)$ belongs to $V$ for every $y$ in $G$, then either $f$ is in $V$ or $m$ is exponential.