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
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 ?