On almost smooth functions and piecewise smooth functions

Piecewise smooth (PS) functions are perhaps the best-known examples of semismooth functions, which play key roles in the solution of nonsmooth equations and nonsmooth optimization. Recently, there have emerged other examples of semismooth functions, including the p-norm function (1 <p <∞) defined on Rⁿ with n ≥ 2, NCP functions, smoothing/penalty functions, and integral functions. These semismooth functions share the special property that their smooth point sets are locally connected around their nonsmooth points. By extending a result of Rockafellar, we show that the smooth point set of a PS function cannot have such a property. This shows that the above functions, though semismooth, are not PS. We call such functions almost smooth (AS). We show that the B-subdifferential of an AS function at a point has either one or infinitely many elements, which contrasts with PS functions whose B-subdifferential at a point has only a finite number of elements. We derive other useful properties of AS functions and sufficient conditions for a function to be AS. These results are then applied to various smoothing/penalty functions and integral functions. © 2006 Elsevier Ltd. All rights reserved.