Can the depicted function be a solution of an ODE with locally Lipschitz autonomous vector field?

Question

Problem: Can x(t) depicted be a solution of a scalar differential equation x(dot)=f with locally Lipschitz autonomous

Answer 1

I find your explanation unsatisfactory, since it makes no use of the Lipschitz condition (and this condition is necessary).

The key points to emphasize:

1. Horizontal stretch means that there is an interval within which $x(t)= x_0$, a constant. This implies $f'(x_0)=0$, i.e., $x_0$ is an equilibrium point.

2. The same ODE admits the constant solution $x(t)\equiv x_0$ (its graph is horizontal line).

3. The locally Lipschitz property of $f$ implies uniqueness for IVP, which means any two solutions that agree at one moment of time have to agree always. But this contradicts the observation made in 2: the shown solution and the constant solution agree in some places and become different elsewhere.