On almost smooth functions and piecewise smooth functions (Q881610)

The paper contains contributions on the study of the class of piecewise smooth functions (PS) and other classes of semismooth functions, \(f:O\to{\mathbb R},\;O\subset {\mathbb R}^n,\) \(O\) 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.