On almost smooth functions and piecewise smooth functions

Liqun Qi, Paul Tseng

Research output: Journal article publicationJournal articleAcademic researchpeer-review

19 Citations (Scopus)


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 Rnwith 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.
Original languageEnglish
Pages (from-to)773-794
Number of pages22
JournalNonlinear Analysis, Theory, Methods and Applications
Issue number3
Publication statusPublished - 1 Aug 2007
Externally publishedYes


  • Newton method
  • Nonsmooth function
  • Piecewise smooth function
  • Semismooth function
  • Subdifferential

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Mathematics(all)


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