Let Alg N be a nest algebra associated with the nest N on a (real or complex) Banach space X. Suppose that there exists a non-trivial idempotent P in Alg N with range P(X) in N , and δ: Alg N --> Alg N is a continuous linear mapping (generalized) left derivable at P, i.e. δ(ab) = aδ(b) + bδ(a) (δ(ab) = aδ(b) + bδ(a) − baδ(I)) for any a; b in Alg N with ab = P, where I is the identity element of Alg N . We show that δ is a (generalized) Jordan left derivation. Moreover, in a strongly operator topology we characterize continuous linear maps δ on some nest algebras Alg N with the property that δ(P) = 2P δ(P) or δ(P) = 2P δ(P) − P δ(I) for every idempotent P in Alg N .
