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][1] 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][2].
Thanks. 

  [1]: https://www.encyclopediaofmath.org/index.php/Osgood_criterion
  [2]: https://math.stackexchange.com/questions/158888/does-log-lipschitz-regularity-imply-h%C3%B6lder-continuity