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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, $f:O\to{\Bbb R},\;O\subset {\Bbb R}^n,$ $O$ 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 $\varepsilon>0$, such that $B_{\overline{\varepsilon}}(x)\cap X_f$ is connected for any $0<\overline{\varepsilon}<\varepsilon$. 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:
26B05Continuity and differentiation questions (several real variables)
26A27Nondifferentiability of functions of one real variable; discontinuous derivatives
26B25Convexity and generalizations (several real variables)
26B35Special properties of functions of several real variables, Hölder conditions, etc.
49J52Nonsmooth analysis (other weak concepts of optimality)
52A41Convex functions and convex programs (convex geometry)
65D15Algorithms for functional approximation
90C25Convex programming
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