Say we have a continuous function (perhaps not everywhere differentiable) that satisfies an ODE $y^\prime(x)=h(y(x),x)$ for almost all $x$ in $[0,1]$.
Are the any references for that deal with basic ODE questions (existence, uniqueness) for these class of solutions? If so which ones you would recommend?
Mere continuity is not enough for viable theory. Indeed, the Cantor function is continuous and satisfies $y'(x)=0$ for almost all $x$. So do linear combinations of its shifts, and a countless number of other similar functions $y$. So we don't have anything like a uniqueness theorem in this context.
The appropriate assumption on $y$ is absolute continuity. This is equivalent to $y$ being an indefinite integral of a Lebesgue integrable function (which agrees with $y'$, of course). We can recast the problem as an integral equation $$y(x) = y_0+ \int_{x_0}^x h(y(t),t)\,dt $$ and perhaps attempt the Picard iteration.
A classical existence result for equations with discontinuous $h$ is Carathéodory's existence theorem. Under additional assumption one can get uniqueness: see the notes On Discontinuous Differential Equations by Bressan and Shen, where further references may be found.