Let A be a Banach algebra, and let E be a Banach A-bimodule. A linear map S:A→E is intertwining if the bilinear map

Formula

is continuous, and a linear map D:A→E is a derivation if δ1D=0, so that a derivation is an intertwining map. Derivations from A to E are not necessarily continuous.

The purpose of the present paper is to prove that the continuity of all intertwining maps from a Banach algebra A into each Banach A-bimodule follows from the fact that all derivations from A into each such bimodule are continuous; this resolves a question left open in [1, p. 36]. Indeed, we prove a somewhat stronger result involving left- (or right-) intertwining maps.