question on lipschitz continuity and boundednes

Question

Can we get an example of a nonlinear vector valued function $f:[t_0, T]\times \mathbb{R}^n\times \mathbb{R}^m\rightarrow \mathbb{R}^n$ which is continuous on its domain and bounded on its domain, but is not Lipschitz continuous on its domain. And also an example of the function which is continuous on its domain, Lipschitz continuous w.r.t second and third arguments on its domain but is unbounded on its domain.

Answer 1

Yes.

As for the first question, assume $t_0=0$ and let $f(x,y,z)= x\sin(\frac{1}{x})e$ where $e$ is any element of the target space.

For the second question consider a linear nonconstant map $g: \mathbb{R}^n\rightarrow \mathbb{R}^n$ and let $f(t,x,y)= g(x)$.