I have to verify the applicability of Picard-Lindelöf theorem for the Cauchy problems associated with the ODE $$x'(t) = |\sin{x(t)}|(t-e^{x(t)}).$$
In order to answer I have to verify that:
$f(x,t)=|\sin{x(t)}|(t-e^{x(t)})$ is $\mathcal{C}(\mathbb{R^2})$
$f(x,t)$ is locally Lipschitz continuous
The first condition is verified. For the second condition I tried to calculate the first order partial derivatives $\frac{\partial f(x,t)}{\partial x}$. If they are continuous, the condition is verified. $$f(x,t)=\cases{\sin{x(t)}(t-e^{x(t)})\quad \sin{x(t)}\ge0\\-\sin{x(t)}(t-e^{x(t)})\quad \sin{x(t)}<0}$$ $$f'(x,t)=\cases{\cos{x(t)}(t-e^{x(t)})-\sin{x(t)}e^{x(t)}\quad \sin{x(t)}\ge0\\-\cos{x(t)}(t-e^{x(t)})+\sin{x(t)}e^{x(t)}\quad \sin{x(t)}<0}$$ but the derivative is not continuous for $\sin{x(t)}=0$ (i.e $\cos{x(t)}=1$ and $x=0+2k\pi$) because $$t\neq-t,$$ so I cannot say that there is an unique solution for the Cauchy problems associated.
Is my approach correct?