Yes. This follows from the classical uniqueness theorem due to Osgood (the original [paper][1] appeared in 1898).

> **Osgood's Criterion.** Let $\omega(t,u)=\phi(t)\psi(u)$ where $\phi(t)\geq 0$ is continuous 
on the interval $(0,a)$ and $\psi(u)$ is continuous on $\mathbb R_{+}$, $\psi(0)=0$, $\psi(u)>0$ for $u>0$, and 
$$\int_{0}^{\epsilon}\phi(t)dt<\infty,\qquad \int_{0}^{\epsilon}\frac{du}{\psi (u)}=\infty$$
for some $\epsilon>0$. Suppose that the mapping $f:[0,a]\times B_R(x_0)\to \mathbb R^d$ satisfies the condition
$$\|f(t,x_1)-f(t,x_2)\|\leq\omega (t,\|x_1-x_2\|)$$
for any $t\in(0,a]$ and any $x_1,x_2\in B_R(x_0)$. Then the initial value problem
$$\dot{x}=f(t,x),\qquad x(0)=x_0$$
has at most  one solution on the interval $[0,\delta]$ with some $\delta>0$.

Osgood's theorem allows for the mappings $f(t,x)$ which are discontinuous at $t=0$. (Actually, the condition that $\phi(t)$ is continuous on $(0,a)$ can be replaced with an assumption
of mere integrability.)  Of course, the existence of a local solution is implied by the Peano theorem under the additional assumption that $f$ is continuous in $(t,x)$.

Moreover, Wintner showed that Osgood's uniqueness condition implies the  convergence of  successive Picard iterations to a local solution on a sufficiently small interval (A. Wintner, ["On the Convergence of Successive Approximations"][2], *Amer. Journal of Math.* Vol. 68 (1946), pp. 13-19).


  [1]: https://doi.org/10.1007/BF01707876 "Osgood, W.F. Beweis der Existenz einer Lösung der Differentialgleichung dy/dx = f(x,y) ohne Hinzunahme der Cauchy-Lipschitz'schen Bedingung. Monatsh. f. Mathematik und Physik 9, 331–345 (1898). zbMATH review at https://zbmath.org/?q=an:29.0260.03"
  [2]: https://www.jstor.org/stable/2371736 "doi:10.2307/2371736. zbMATH review at https://zbmath.org/?q=an:0063.08288"