Given an even function how to obtain the most close odd function and vise versa?
Why it was closed?
I hadn't seen the question until just now, nor have I tried to follow what you're asking mathematically. But the comments express some pretty clear objections. "Off-topic" seems like an inappropriate closure reason, but then, the list of options doesn't always contain what we want.
As Scott says, it's not helpful that you've deleted your own comments, because (for instance) it makes it harder to answer your question above.
Let me also agree with Yemon: you need to clarify what you mean by "function", otherwise the question isn't precise. I suppose you mean something like "Lebesgue-integrable function from $\mathbb{R}$ to $\mathbb{R}$", but don't make us guess: tell us!