# 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 $$varepsilon to 0$$?