  We prove that every weak-local derivation on a C$^*$-algebra is continuous,
and the same conclusion remains valid for weak$^*$-local derivations on von
Neumann algebras. We further show that weak-local derivations on C$^*$-algebras
and weak$^*$-local derivations on von Neumann algebras are derivations. We also
study the connections between bilocal derivations and bilocal $^*$-automorphism
with our notions of extreme-strong-local derivations and automorphisms.
