Automatic Continuity of $\sigma$-Derivations on $C^*$-Algebras

Let $A$ be a $C^*$-algebra acting on a Hilbert space $H$, $\sigma:A\to B(H)$ be a linear mapping and $d:A\to B(H)$ be a $\sigma$-derivation. Generalizing the celebrated theorem of Sakai, we prove that if $\sigma$ is a continuous $*$-mapping then $d$ is automatically continuous. In addition, we show the converse is true in the sense that if $d$ is a continuous $*-\sigma$-derivation then there exists a continuous linear mapping $\Sigma:A\to B(H)$ such that $d$ is $*-\Sigma$-derivation. The continuity of the so-called $*-(\sigma,\tau)$-derivations is also discussed.