Hello!

Can anyone help me solve the problems? It paralyzes my work

It is obvious in the first two problems that and is Lipschitz. But I can't go any further.

1) A continuously differentiable function is defined on such that and for all . Show that is constant.

2) Suppose that for some positive constant , , . Show that if , then for any , .

3) Show that there exists no real-valued function such that and for all .

Thank you very much!