The paper contains contributions on the study of the class of piecewise smooth functions (PS) and other classes of semismooth functions,

$f:O\to \mathbb{R},\phantom{\rule{0.277778em}{0ex}}O\subset {\mathbb{R}}^{n},$ $O$ 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

$\epsilon >0$, 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.