  We prove new results on generalized derivations on C$^*$-algebras. By
considering the triple product $\{a,b,c\} =2^{-1} (a b^* c + c b^* a)$, we
introduce the study of linear maps which are triple derivations or triple
homomorphisms at a point. We prove that a continuous linear $T$ map on a unital
C$^*$-algebra is a generalized derivation whenever it is a triple derivation at
the unit element. If we additionally assume $T(1)=0,$ then $T$ is a
$^*$-derivation and a triple derivation. Furthermore, a continuous linear map
on a unital C$^*$-algebra which is a triple derivation at the unit element is a
triple derivation. Similar conclusions are obtained for continuous linear maps
which are derivations or triple derivations at zero.
  We also give an automatic continuity result, that is, we show that
generalized derivations on a von Neumann algebra and linear maps on a von
Neumann algebra which are derivations or triple derivations at zero are all
continuous even if not assumed a priori to be so.
