The author of the present pleasant paper establishes that if $A$ is a subalgebra of a Banach algebra $B$ and $f: A \to B$ satisfies $\Vert f(x+y) - f(x) - f(y)\Vert \leq \delta$ and $\Vert f(xy) - xf(y) - f(x)y\Vert \leq \varepsilon$, for all $x, y \in A$ and for some $\delta, \varepsilon \geq 0$, then there exists a unique additive derivation $d:A \to B$ such that $\Vert f(x) - d(x)\Vert \leq \delta \quad (x \in A)$, and $x\left(f(y) - d(y)\right) = 0 \quad (x, y \in A)$. The result and its proof are still true for a more general case if we consider a normed algebra $A$ and replace $B$ by a Banach $A$-bimodule $X$.
He also proves that if $B$ is a normed algebra with an identity belonging to $A$, then every mapping $f: A \to B$ satisfying $\Vert f(xy) - xf(y) - f(x)y\Vert \leq \varepsilon \quad (x, y \in A)$ must fulfil $f(xy) = xf(y) - f(x)y \quad (x, y \in A)$. This superstability result is nice, since there is not assumed any (approximately) additive condition on $f$. Some similar results in which one considers generalized derivations can be found in the reviewer’s paper [“Hyers-Ulam-Rassias stability of generalized derivations”, Int. J. Math. Math. Sci. (to appear)].