This paper presents a detailed duality theory for a class of continuous linear programs called separated continuous linear programs (SCLP), based on a particular dual problem SCLP*. Using weak duality, a notion of complementary slackness is introduced, and several sufficient conditions for optimality of SCLP are derived along with the existence of complementary slack variables for basic feasible solutions for SCLP. Following this, a fairly general condition for the absence of a duality gap between SCLP and SCLP* is given, as are several conditions for the existence of an optimal solution for SCLP*. Finally, using all these ingredients, a strong duality result between SCLP and SCLP* is proven when the problem data are piecewise analytic. A simple counterexample is presented to show that strong duality may not follow if the assumptions of piecewise analyticity do not hold.