My math for engineering teacher has made a claim using the Cauchy-Lipschitz theorem that I fail to justify using other sources. Please keep in mind that my question relates the Cauchy-Lipschitz theorem in the context of differential equations defined on intervals of $\mathbb{R}$ as I seem to understand this theorem may be applicable to broader subjects.
For what I understand the Cauchy-Lipschitz theorem claims that for $(E)$ such that : $$ (E) : \left\{ \begin{array}{ccc} \frac{dy}{dt} &=& f(t, y)\\ y(x_0) &=& y_0\\ \end{array} \right. $$
with $y \in C_1$ and $y$ defined on some interval of $A \subset\mathbb{R}$. Let, for readability purposes, define $L$ to be the set of all local Lipschitz functions with respect to the second variable. Then
$$ ( \forall t \in A, f \in L) \Rightarrow \exists!\ \ g,\ \ g\texttt{ satisfy } (E). $$
My teacher then uses this theorem in a way I fail to grasp : in one of their exercises they claim that the following differential equation defined for $t \in [0, +\infty]$ :
$$ (E') : \left\{ \begin{array}{ccc} \frac{dy}{dt} &=& 2\sqrt{|y|}\\ y(0) &=& 0\\ \end{array} \right. $$
has infinitely many solutions because $2\sqrt{|y|}$ is not a Lipschitz function with respect to $y$ for $t\rightarrow 0$. This is where I'm getting confused because it seems to me that they mean
$$ ( \forall t \in A, f \not\in L) \Rightarrow \exists^{\infty}\ \ g,\ \ g\texttt{ satisfy } (E), $$
which doesn't fit my understanding of the Cauchy Lipschitz theorem. Would you mind help me clear the confusion ?