Let us consider a bounded, Borel vector field $F\colon (0,T) \times \mathbb R^d \to \mathbb R^d$ for some $T>0$. Assume it satisfies the following Osgood-like condition: $$\tag{O} \boxed{\langle F(t,x) - F(t,y), x-y \rangle \le C(t) \Vert x-y \Vert \rho( \Vert x-y \Vert)} \qquad \forall x,y \in \mathbb R^d, \, \forall t \in (0,T), $$ where $C\in L^1(0,T)$ and $\rho \colon [0,1) \to \mathbb [0,+\infty) $ is an Osgood modulus of continuity, i.e. a continuous, non-decreasing function with $\rho(0)=0$ and $$ \int_0^1 \frac{1}{\rho(s)} \, ds = +\infty. $$

Q. Is it true that $F$ is continuous? Or, more precisely, is it true that $F$ is equivalent, up to null sets, to a continuous function?

Beside the link above, something (mildly) related can also be found here. Thanks.