Inverse function Theorem

[cap.r]

1. The problem statement, all variables and given/known data
Give an example of a continuously differentiable mapping F:R^n --> R^n with the property that tehre is no open subset U of R^n for which F(U) is open in R^n


2. Relevant equations
let U be an open subset of R^n and supposed that the continuously differentiable mapping F:U-->R^n is stable and has an invertible derivative matrix at each point. Then it's image F(u) is also open.


3. The attempt at a solution

So from the theorem I stated it seems like if F is not stable at any point then F(U) is not open. so I just need to give a function whose Jacobian is non-invertible. i can just think of X^2 which isn't one to one. but that's in R^2 and this is asking for an example in R^n...

[lanedance]

X^2 though not being 1:1, will still take an open set to an open set... and unless i'm understanding wrong, it is from R -> R

if I read you theorem correctly, the problem suggest F does not have an invertible derivative matrix & so is not 1:1

so have a think about closed sets, is an isolated point a closed set?

[lanedance]

updated

[cap.r]

ok so if I let the mapping F be degenerative and map everything to a k in R^n. then it will take the open set U to a closed set in R^n. so thanks for the help i got it.