Existence of solutions for implicit functional-integral equations associated with discontinuous functions

Let $I:=[0,1]$. In this paper we deal with the functional-integral equation $$h(u(t))=g(t)+f(t,int_Ik(t,s),u(arphi(s)),ds)quadhbox{for a.e.}quad tin I.$$ While in most literature the function $f$ is usually requested to be a Carath'eodory map, we prove an existence result for solutions $uin L^p(I)$ (with $pin]1,+infty]$), where the continuity of $f$ with respect to the second variable is not assumed. As a matter of fact, a function $f$ satisfying the assumptions of our result can be discontinuous, with respect to the second variable, even at each point of its domain.