In this paper it is shown that
derivations from L1[0,1] into Banach-L1[0,1]-modules are automatically
continuous. The result is obtained as a corollary of a theorem in which sufficient
conditions on the closed ideals of a separable commutative Banach algebra B are
given so that every module derivation from B into a Banach-B-bimodule is
continuous. One of the conditions obtained is best possible. For the general case of
a Banach algebra A (not necessarily commutative or separable) sufficient
conditions on the closed ideals are also given to force the continuity of module
derlvations and of certain higher derivations from any Banach algebra onto
A.
