The M-components of level sets of continuous functions in WBV

We prove that the topographic map structure of upper semicontinuous functions, defined in terms of classical connected components of its level sets, and of functions of bounded variation (or a generalization, the WBV functions), defined in terms of M-connected components of its level sets, coincides when the function is a continuous function in WBV. Both function spaces are frequently used as models for images. Thus, if the domain [omega] of the image is Jordan domain, a rectangle, for instance, and the image u [member of] C([omega]) [intersection] WBV([omega]) (being constant near [delta omega]), we prove that for almost all levels [lambda] of u, the classical connected components of positive measure of[u [greater than or equal] [lambda]] coincide with the M-components of [u [greater than or equal] [lambda]]. Thus the notion of M-component can be seen as a relaxation of the classical notion of connected component when going from C([omega]) to WBV([omega])