interval either. At the same time, the function is of bounded variation on the interval . However, all three functions are of bounded variation on each interval with . Cantor function is a well-known example of a function of bounded variation that is not absolutely continuous. The Sobolev space is a proper subset of . In fact, for each in it is possible to choose a measure (where is the Lebesgue measure on ) such that the equality holds, since it is nothing more than the definition of weak derivative, and hence holds true. One can easily find an example of a BV function which is not : in dimension one, any step function with a non-trivial jump will do. Applications Mathematics Functions of bounded variation have been studied in connection with the set of discontinuities of functions and differentiability of real functions, and the following results are well-known. If is a real function of bounded variation on an interval then is continuous except at most on a countable set; has one-sided limits everywhere (limits from the left everywhere in , and from the right everywhere in ; the derivative exists almost everywhere (i.e. except for a set of measure zero). For real functions of several real variables the characteristic function of a Caccioppoli set is a BV function: BV functions lie at the basis of the modern theory of perimeters. Minimal surfaces are graphs of BV functions: in this context, see reference . Physics and engineering The ability of BV functions to deal with discontinuities has made their use widespread in the applied sciences: solutions of problems in mechanics, physics, chemical kinetics are very often representable by functions of bounded variation. The book details a very ample set of mathematical physics applications of BV functions. Also there is some modern application which deserves a brief description. The Mumford–Shah functional: the segmentation problem for a two-dimensional image, i.e. the problem of faithful reproduction of contours and grey scales is equivalent to the minimization of such functional. Total variation denoising See also Renato Caccioppoli Caccioppoli set Lamberto Cesari Ennio de Giorgi Helly's selection theorem Locally integrable function Lp(Ω) space Lebesgue–Stieltjes integral Radon measure Reduced derivative Riemann–Stieltjes integral Total variation Aizik Isaakovich Vol'pert Total variation denoising Notes References Research works . . Includes a discussion of the functional-analytic properties of spaces of functions of bounded variation. . , particularly part I, chapter