Multiple choice question on Inverse Function Theorem

by ASHWINI SANKHE   Last Updated March 13, 2018 07:20 AM

Let $$f:\Bbb R^n \to \Bbb R^n$$ be a continuously differentiable map satisfying $\vert \vert f(x)-f(y) \vert \vert \ge \vert \vert x-y \vert \vert $ for all $x,y \in \Bbb R^n$ Then,

1) f is onto.

2) $f(\Bbb R^n)$ is closed subset of $\Bbb R^n$.

3) $f(\Bbb R^n)$ is open subset of $\Bbb R^n$.

4)$f(0)=0$.
By inverse function theorem,$f'(x)\neq 0 $ as $\vert \vert f(x)-f(y) \vert \vert \ge \vert \vert x-y \vert \vert $ for all $x,y \in \Bbb R^n$
Hence $f'(X)$ is locally invertible and $\Bbb R^n $ is both open and closed so correct answers are options 1,2,3.Is this correct justification?

Tags : real-analysis


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