real analysis – Approximating the initial value problem $x'(t)=sqrt{|x(t)|}, x(0)=0$ by solutions of ODEs with smooth right hand side

Consider the initial value problem $$x'(t) = sqrt{|x(t)|}, x(0) = 0.$$ Note that the right hand side is not Lipschitz at $x=0$. There are infinitely many solutions to this, e.g. $x_0equiv 0$ and for any $r > 0$, $$x_r(t) = begin{cases} 0 &tleq r,\ frac{1}{4} (t – r)^2 & t > r.end{cases}$$ My question is: If we pick one of these solutions, call it $x$, can we find a $C^0_{loc}$-approximation of $f(x):=sqrt{|x|}$ by $C^{infty}$-functions $f_{varepsilon}(x)$ such that the solutions of an appropriate initial value problem (which are then necessarily unique) for the ODEs $x_{varepsilon}'(t) = f_{varepsilon}(x_{varepsilon}(t))$ converge to $x$ as $varepsilon to 0$?