Weak-local derivations and homomorphisms on C*-algebras

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