determinant not zero

[Someonekicked]

Suppose f: R^n -> R^n is continuously differentiable in an open set V
containing a, and det f'(a) != 0.
show that there exists a closed rectangle U containing a in its interior
such that det f'(x) != 0 for all x in U.

im not sure where to begin cuz of the determinant.

just an idea, even though it looks to me wrong,
Since f is continuously differentiable, then f'(x) exists for x in V, and
f'(x) is continuous? (can we say a matrix is continuous?)
Since f'(x) is continuous, then there exists d > 0 such that
|x - a| < d => | det( f'(x) - f'(a) ) | < | det( f'(a) ) |
then using something similar to the triangle inequality to
prove det f'(x) is positive!?

any suggestions or hints would be appreciated!

thx in avdance.

--
Quotes from The Weather Man:
Robert Spritz: Do you know that the harder thing to do, and the right thing
to do, are usually the same thing? "Easy" doesn't enter into grown-up
life... to get anything of value, you have to sacrifice.

[Peter L. Montgomery]

In article <kaOdndI4ZaX...@comcast.com>

"Someonekicked" <someon...@comcast.net> writes:
>Suppose f: R^n -> R^n is continuously differentiable in an open set V
>containing a, and det f'(a) != 0.
>show that there exists a closed rectangle U containing a in its interior
>such that det f'(x) != 0 for all x in U.
>
>im not sure where to begin cuz of the determinant.
>
>just an idea, even though it looks to me wrong,
>Since f is continuously differentiable, then f'(x) exists for x in V, and
>f'(x) is continuous? (can we say a matrix is continuous?)

f'(x) being continuous (at a) means each of the n^2 partial
derivatives is continuous. The matrix is continuous
if each entry is continuous. The determinant is
a polynomial function of the matrix entries.

>Since f'(x) is continuous, then there exists d > 0 such that
>|x - a| < d => | det( f'(x) - f'(a) ) | < | det( f'(a) ) |
> then using something similar to the triangle inequality to
>prove det f'(x) is positive!?
>
>any suggestions or hints would be appreciated!

--
The USA must invest in interstate passenger rail. Amtrak needs high-speed
public railways, not private lines where it waits for freight trains to pass.

pmon...@cwi.nl Microsoft Research and CWI Home: Bellevue, WA

[Someonekicked]

thx that helps!

--
Quotes from The Weather Man:
Robert Spritz: Do you know that the harder thing to do, and the right thing
to do, are usually the same thing? "Easy" doesn't enter into grown-up
life... to get anything of value, you have to sacrifice.

"Peter L. Montgomery" <Peter-Lawren...@cwi.nl> wrote in message
news:It5z3...@cwi.nl...