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.
question on lipschitz continuity and boundednes
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continuity
lipschitz-functions
1 Answers
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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 $f(t,x,y)= g(x)$.
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0But i need an example for nonlinear function f – 2017-02-14
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0@Vijay This you did not write. Add any bounded nonlinear uniformly $C^1$ function. – 2017-02-14
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0can you give an example? – 2017-02-14
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0@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. – 2017-02-14