On a counterexample in connection with the Picard-Lindel\"of theorem

We give an example, which demonstrates that in the situation of the Picard-Lindel\"of theorem, the Lipschitz condition on the right hand side $f(x,y)$ with respect to $y$, cannot be replaced by Lipschitz continuity in $y$ for every $x$. We show that, in our example, the classical Euler method detects only one of infinitely many solutions and we outline how the latter can be adjusted to find also other solutions numerically.