Consider the initial value problem $ \left\{\begin{array}{ll} u'(t)= 2 \sqrt{|u(t)|}, & \\ u(t_0)=u_0 \in \mathbb{R}. & \end{array}\right. $
Why is $f: [0,T] \times \{v \in \mathbb{R} \mid |v-\frac{1}{4}|\leq r\}$ with $r>0$ defined by $f(t,v):=2 \sqrt{|v|}$ uniformly lipschitz continuous in v? I want to use Picard-Lindelöf here.