# question on lipschitz continuity and boundednes

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.

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 $f(t,x,y)= g(x)$.
• @Vijay This you did not write. Add any bounded nonlinear uniformly $C^1$ function. – Thomas Feb 14 '17 at 5:36
• @Vijay just add $\exp(-||z||^2)$ or $\exp(-||x||^2)$. This is a smooth bounded function with bounded derivatives, so also with bounded Lipshitz constant. – Thomas Feb 14 '17 at 15:49