Here's my answer.Let be an odd regulated function. Prove that:

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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:

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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 .

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We can use the fact is odd:

We can then rewrite this as:

Is this the right?