Sign-decay differential equation
= Sign-decay differential equation
For $y_0>0$, the initial-value problem
$$
y'=-\operatorname{sign}(y),
\qquad y(0)=y_0,
$$
has the continuous piecewise differentiable solution
$$
y(t)=\max\{y_0-t,0\}.
$$
The vector field is not Lipschitz at zero, but the direction on each side prevents any solution from leaving zero, so this solution is unique in the piecewise differentiable class.