Confused on Uniqueness and Existence of ODEs

Question

We have that $|t|\frac{dy}{dx}=\sqrt{|y|}$ with $y(t_0)=y_0$ and I am asked to state the values of $y_0$ and $t_0$ such that we are guaranteed a solution exists. Now I understand if $f(y,t)=\frac{\sqrt{|y|}}{|t|}$ is continuous on an interval containing $t_0$ then we have a solution, but $y(t)=0$ with $y(0)=0$ is a solution to the differential equation but $f(y,t)$ is clearly not continuous at $t=0$, so what in what I have said/understanding is wrong, or

Answer 1

The existence theorem is not an " if and only if" theorem.

If $f(t,y)$ is continuous then we have a solution for $ y'=f(t,y)$ but the continuity is not necessary for having a solution.