- #1
boombaby
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Homework Statement
f:Rn->Rn is continuous and satisfies
|f(x)-f(y)|>=k|x-y|
for all x, y in Rn and some k>0. Show that F has a continuous inverse.
Homework Equations
The Attempt at a Solution
It is easy to show that f is injective, but I've no idea how to prove the surjectivity. I was thinking on the R1->R1 case for a while, and guess that I can show that f is unbounded to deduce the surjectivity. But it seems that boundedness is not that useful in Rn->Rn case.
Any hint? THanks!