A ring \(R\) is said to be clean if every element of \(R\) is a sum of an idempotent and a unit. These rings were introduced by \textit{W. K. Nicholson} [Trans. Am. Math. Soc. 229, 269-278 (1977; Zbl 0352.16006)] in relation with the study of the exchange property, and it is known that the class of clean rings is properly contained in the one of exchange rings. One of the motivations for this paper is a result of M. Ó\,Searcóid according to which the ring of linear transformations of a vector space is clean. Another motivation is a result of S. H. Mohamed and B. J. Müller which says that continuous modules satisfy the exchange property and can be reformulated as: the endomorphism ring of a continuous module is an exchange ring. For brevity, the authors call a module clean whenever its endomorphism ring is clean, and the main result of the paper extends the two results just mentioned by showing that each continuous module is clean. The authors also obtain some nice consequences of this theorem. For example, they prove that every quasi-continuous module with the (finite) exchange property is clean and also that every quasi-projective right module over a right perfect ring is clean.