Question about inverse function

  • I
  • Thread starter facenian
  • Start date
  • #1
394
15
Let ##f:U\subset R^n\rightarrow V\subset R^n## be a biyective function of class ##C^m(m\geq 1)##, ##U## and ##V## are open sets in ##R^n##. I know from the inverse funtion theorem that when ##J(f)\neq 0## in a point of the the domain a local inverse exists, however, given the above conditions I'd like to know if it is true that ##J(f)\neq 0## in U.(kind of a reciprocal of the inverse function theorem)
For instance, if the inverse of a differentiable function is differentiable this would be easy to prove since
$$\frac{\partial(x_1\ldots x_n)}{\partial(y_1\ldots y_n)}\,\frac{\partial(y_1\ldots y_n)}{\partial(x_1\ldots x_n)}=1\neq 0$$
However I don't know whether this is correct or not.
 

Answers and Replies

  • #2
mathwonk
Science Advisor
Homework Helper
2020 Award
11,100
1,302
x^3.
 
  • Like
Likes member 587159 and FactChecker

Related Threads on Question about inverse function

  • Last Post
Replies
3
Views
1K
Replies
17
Views
5K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
2
Views
860
  • Last Post
Replies
10
Views
2K
Replies
2
Views
2K
Replies
3
Views
1K
Replies
2
Views
1K
Top