Summary: Given \(T>0\), a set \(Y\subseteq\mathbb{R}^n\), a point \(\xi\in\mathbb{R}^n\) and two functions \(f:[0,T]\times\mathbb{R}^n\to\mathbb{R}\) and \(g:Y\to\mathbb{R}\), we are interested in the Cauchy problem \(g(u')=f(t,u)\) 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 \(x\in\mathbb{R}^n\). 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 \(\mathcal{S}(\xi)\) admits an upper semicontinuous and compact-valued multivalued selection.