Suppose that $\frak A$ is a $C^*$-algebra acting
on a Hilbert space $\frak K$, and $\varphi, \psi$ are mappings
from $\frak A$ into $B(\frak K)$ which are not assumed to be
necessarily linear or continuous. A $(\varphi, \psi)$-derivation
is a linear mapping $d: \frak A \to B(\frak K)$ such that
$$d(ab)=\varphi(a)d(b)+d(a)\psi(b)\quad (a,b\in \frak A).$$ We prove that if $\varphi$ is a multiplicative (not
necessarily linear)\ $*$-mapping, then every
$*$-$(\varphi,\varphi)$-derivation is automatically continuous.
Using this fact, we show that every
$*$-$(\varphi,\psi)$-derivation $d$ from $\frak A$ into
$B(\frak K)$ is continuous if and only if the $*$-mappings
$\varphi$ and $\psi$ are left and right $d$-continuous,
respectively.