1. Introduction. In [6] and [8] we showed that centralisers (that is operators commuting with multiplication) on certain Banach algebras are necessarily continuous. In the present paper we consider whether an operator on a Banach space commuting with a continuous or a closed operator must itself be continuous. We also consider the more general problem of whether an operator S; E1 -- E2, satisfying ST1 = T2S, with T1, T2 continuous, must be continuous. If the operators T1, T2 are operators of scalar type [3, p. 226] or operators in BK spaces [15, p. 29] given by triangular matrices, we see that the only discontinuous S are the obvious ones. In ?8 we consider operators in L1(-oo, + oo) and le(-oo, + oo) commuting with the translation operator (Tf)(x) =f(x + 1). For the L1 space any operator commuting with T is continuous, whereas in W6(-oo, +oo) we can find a discontinuous operator commuting with all the translation operators. A full solution to this problem might be useful in discussing the extent to which discontinuous homomorphisms between Banach algebras exist since if 0 is a homomorphism it satisfies f(ab)=0(a)0(b) which can be written ?bLa=L<(a) where LaX = ax. 2. Discontinuous commutants. The Banach-Steinhaus theorem [2, p. 27] makes it necessary to employ transfinite methods to "construct" discontinuous operators in Banach spaces. Thus it can be difficult, when considering whether there are discontinuous operators satisfying certain conditions, to show that the answer is affirmative. We consider two cases in which the construction is simple. First, however, some notation.