I believe that if one considers the ODE $$ x'(t)=f(t,x),\qquad x(0)=x_0 $$ where one assumes that $f(t,x)\in L^{\infty}(\mathbb{R}^2)$ and $f$ is Lipschitz with respect to $x$, then there exists a unique Lipschitz solution to the ODE $x(t)\in C^{0,1}(-\epsilon,\epsilon)$ for $\epsilon$ sufficiently small. Here the differential equation is understood to be equality almost everywhere (using Rademacher's Theorem for instance). Indeed, one can prove this using the standard proof by rewriting the problem as $$ x(t)=x_0+\int_0^t f(t',x(t'))dt'. $$ The only difference between this and the standard treatment, is that we do not assume that $f$ is continuous with respect to $t$ (only $L^{\infty}$), and as a result, we do not get that $x(t)$ is $C^1$ (merely Lipschitz).
I believe this is correct, but wanted to ask if there are any errors? This seems like a somewhat more natural statement (everything is at the $L^{\infty}$ scale) and I am surprised that I wasn't able to find any references to it.
