Hi

I have to prove the following theorem from my analysis book.

If

and

are both defined on

and

is odd, show that

is also odd. Is the statement

still true if

is replaced by any symmetric interval

, with

?

Following is my attempt. Since both f and its inverse are defined on R ,

the function f is onto and off course one-to-one.

Let

be arbitrary.

and since

is also defined, we have

. But since

is the domain of

,

.

or

The last step follows since f is an odd function.

But we had assumed that

So it follows that

So this proves the first part. For the second part, I will argue that since its

a symmetric interval , whenever

and the same reasoning will follow.

Can you comment on the proof ?

thanks

newton