Isn't for all odd functions? Why do you need it to be regulated?
Here's my answer.Let be an odd regulated function. Prove that:
.
By definition of a regulated function where is a sequence of step functions converging uniformly to f.
Claim: is odd.
We already know that . (*)
We can multiply both sides by -1 to get for some .
Since (*) is valid , we also know that:
.
But we can rewrite the first one as since we know that f is odd.
We can now equate the two to get:
So we know that is odd.
My next claim is:
is an odd step function so where .
.
We can use the fact is odd:
We can then rewrite this as:
Is this the right?
That's actually very true! I've come up with something else:
Claim: is odd.
By definition of a regulated function, we can choose , as defined above, s.t where .
Therefore:
(*)
Since , and and are defined at all values of :
But since f is odd:
(**)
So we can add (*) and (**) together to get:
and this implies that or, in a more conventional way,