"First of all, the sum of g(x)+h(x) is a function of odd or even numbers.

That is to say, f(x) cannot be a odd function or an even function. So f(x) is not arbitrary? "

Advanced Mathematics, Tongji Fifth Edition. /kloc-page 0/6 didn't say that!

Only the domain of f(x) needs to be symmetric about the origin. It is arbitrary within this range. Can be an odd or even function.

This only means that any function with a symmetric domain about the origin can be written as the sum of an even function and a odd function. Of course, domains should be consistent and universal.