Suppose the functions $h(t,x)$ and $f(t,x,z,\epsilon)$ and their partial derivatives up to the second order are bounded in some compact domains that contain the origin, where $\epsilon \in [0,\epsilon ^∗)$, $\epsilon^*<<1$. Also suppose that $h(t,0)=0$ and $f(t,0,0,\epsilon)=0$ (that is both of these functions are zero at the origin).
I want to show that $$\Vert\frac{\partial h(t,x)}{\partial t} \Vert \leq k_1 \Vert x\Vert,$$ $$\|f(t,x,z,\epsilon)-f(t,x,z,0)\|\leq \epsilon k_2 (\|x\|+\|z\|).$$
where $k_1, k_2>0$. I read these in a control system textbook, but cannot prove them. I think the first inequality guarantees some sort of smooth behaviour for $h$, but I don't know why the right hand side is a function of $\|x\|$.
The second inequality seems to say that since $f$ is zero at the origin, it is locally Lipschitzs in $\epsilon$. I don't understand this, although it is clear that $f$ is locally Lipschitz in $x$ and $y$ as it's partial derivatives with respect to these arguments are bounded.
