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user193319
user193319
17:45
Problem: Suppose that the mapping $f : \Bbb{R}^n \to \Bbb{R}^n$ is a contraction. Define $g(x) = x - f(x)$ for all $x \in \Bbb{R}^n$. Show that the mapping $g : \Bbb{R}^n \to \Bbb{R}^n$ is both one-to-one and onto. Also show that $g$ and its inverse are continuous.
Okay. The only part of the problem I am having trouble with is showing that $g^{-1}$ is continuous, which is equivalent to showing that $g$ is an open map. I've seen some solutions invoke this idea of Invariance of the Domain, but I don't believe this has been discussed in my book yet.

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