# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On almost smooth functions and piecewise smooth functions. (English) Zbl 1125.26019
The paper contains contributions on the study of the class of piecewise smooth functions (PS) and other classes of semismooth functions, $f:O\to ℝ,\phantom{\rule{0.277778em}{0ex}}O\subset {ℝ}^{n},$ $O$ open. Denote by ${X}_{f}$ the set of smooth points of $f$. One of the obtained main results is the following: if $f$ is a PS function, then ${X}_{f}$ is not locally connected around a point $x\in O\setminus {X}_{f}$. Using this criteria one obtains that a large class of semismooth functions, like the $p$-norms functions, NCP functions, smoothing/penalty and integral functions are not PS functions. In connections with this property the authors introduced the concept of almost smooth functions (AS), namely a function $f$ is AS function if for any $x\in O\setminus {X}_{f}$, there is $\epsilon >0$, such that ${B}_{\overline{\epsilon }}\left(x\right)\cap {X}_{f}$ is connected for any $0<\overline{\epsilon }<\epsilon$. In addition there are introduced some variants of AS functions. A discussion, completed by many examples, about the relationships between these notions and the above classes of semismooth functions is made.
##### MSC:
 26B05 Continuity and differentiation questions (several real variables) 26A27 Nondifferentiability of functions of one real variable; discontinuous derivatives 26B25 Convexity and generalizations (several real variables) 26B35 Special properties of functions of several real variables, Hölder conditions, etc. 49J52 Nonsmooth analysis (other weak concepts of optimality) 52A41 Convex functions and convex programs (convex geometry) 65D15 Algorithms for functional approximation 90C25 Convex programming