It is classically known that complete flat (that is, zero Gaussian curvature) surfaces in Euclidean 3-space R3 are cylinders over
space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a
flat surface f admits singularities but its Gauss map is globally defined on the surface and can be smoothly extended across the
singular set, f is called a frontal. In addition, if the pair (f, ) defines an immersion into R3×S2, f is called a front. A front f is
called flat if the Gauss map degenerates everywhere. The parallel surfaces and the caustic (i.e. focal surface) of a flat front f are
also flat fronts. In this paper, we generalize the classical notion of completeness to flat fronts, and give a representation formula for a
flat front which has a non-empty compact singular set and whose ends are all immersed and complete. As an application, we show
that such a flat front has properly embedded ends if and only if its Gauss map image is a convex curve. Moreover, we show the
existence of at least four singular points other than cuspidal edges on such a flat front with embedded ends, which is a variant of the
classical four vertex theorem for convex plane curves.