Consider the following ODE:
$$\dot{x}=f(x,u),$$ where $x \in \mathbb{R}^n, \; u \in U \subset \mathbb{R}$ compact, and $f \in C^1$ with respect to both $(x,u)$.
Let $\Xi \in Lip_\alpha(\mathbb{R})$ and define $z \doteq x - \Xi(u)$. Assuming $u=u_0 \in U$ fixed, then $$\dot{z} = f(z+\Xi(u_0),u_0) \doteq \tilde{f}(z,u_0).$$
Is the function $\tilde{f}$ defined in this way Lipschitz? Is it $C^1$?
Intuitively I would say $C^1$ with respect to $z$ and Lipschitz with respect to $u_0$, but I am not sure.