Questions related to homeomorphisms of R^n

[math771]

Hey there,

1) The generalization of the Lebesgue theorem is not necessarily true. There are counterexamples where a homeomorphism is not differentiable almost everywhere, such as the Cantor function on the unit interval.

2) The composition of two almost everywhere differentiable homeomorphisms may not be differentiable almost everywhere. This can also be seen with the Cantor function example, where the composition with itself is not differentiable almost everywhere.

3) The inverse of a differentiable almost everywhere homeomorphism may also not be differentiable almost everywhere. The same example with the Cantor function can be used here as well.

Hope that helps!

 

[coquelicot]

Hello,

I have 3 questions related to homeomorphisms ℝ^n to ℝ^n:

1) Is the following generalization of the Lebesgue theorem true?
Every homeomorphism ℝ^n to ℝ^n is differentiable almost everywhere.

2) Assuming the answer to 1) is negative, is the composition of two
almost everywhere differentiable homeomorphisms differentiable almost everywhere?

3) Assuming the answer to 1) is negative, is the invert of a differentiable almost everywhere homeomorphism differentiable almost everywhere?

thx.

 

[fresh_42]

No.
Yes.
I guess so, but am not sure.