I learned the upper-hemicontinuity and L-continuity of set-valued function. Are there a definition of smoothness or differentiability of a set-valued function
$$f:\mathbb R\rightrightarrows\mathbb R?$$
(Let's think about one dimensional case at first.)
For example, can we say that, $f$ is smooth if $f(x)=\cup_{i=1}^{\infty}\{f_i(x)\}$ where $f_i:\mathbb R\to\mathbb R$ is a family of smooth functions?