- #1

- 152

- 0

## Homework Statement

I have some problems with understanding these two things.

Homeomoprhism is a function f [tex] f: M\rightarrow N[/tex] is a homeomorphism if if is bijective and invertible and if both [tex] f, f^{-1}[/tex] are continuous.

Here comes an example, lets take function

[tex] f(x) = x^{3}[/tex] it is clear that f(x) is continuous and bijective and at the same [tex] f^{-1}(x)\rightarrow x^{\frac{1}{3}}[/tex] is also continuous which indicates that f is a homeomorphism. No strange things here

Diffeomorphism- given two manifolds M, N and a bijective map f is called a diffeomorphism if both

[tex] f: M\Rightarrow N[/tex]

[tex] f^{-1}: N\Rightarrow M[/tex] are of class [tex] C^{\infty}[/tex] (if these functions have derivatives of all orders) f is called diffeomorphism

Going back to my example.

f is clearly a homeomorphism but :

why is [tex] f(x)=x^{3}[/tex] of a class [tex] C^{\infty}[/tex]?? what do they mean by 'all orders'

[tex] \frac{df}{dx}=3x^{2}[/tex]

[tex] \frac{d^2f}{dx^2}=6x [/tex]

[tex]\frac{d^3f}{dx^3}=6[/tex]

[tex]\frac{d^4f}{dx^4}=0...[/tex]

and its inversion

[tex] f^{-1}=x^{\frac{1}{3}}[/tex] is not of a [tex] C^{\infty}[/tex]....because

[tex] \frac{df^{-1}}{dx}=\frac{1}{3}x^{\frac{-2}{3}}[/tex]

it is not definied at x=0