On the Cauchy problem for implicit differential equations with discontinuous right-hand side

Given $T>0$, a set $Ysubseteq R^n$, a point $\xiin R^n$ and two functions $f:[0,T] imes R^n o R$ and $g:Y o R$, we are interested in the Cauchy problem $g(u^prime)=f(t,u)$ in $[0,T]$, $u(0)=\xi$. We prove an existence result for generalized solutions of the above problem, where the continuity of $f$ with respect to the second variable is not assumed. As a matter of fact, a function $f(t,x)$ satisfying our assumptions could be discontinuous (with respect to $x$) even at all points $xin R^n$. As regards $g$, we only require that it is continuous and locally nonconstant. We also investigate the dependence of the solution set from the initial point $\xi$. In particular, we give conditions under which the solution multifunction ${cal S}(\xi)$ admits an upper semicontinuous and compact-valued multivalued selection.