Just define the even part of f(x) to be , and the odd part of f(x) to be .
First, it is easy to show that E(x) is an even function. For this amounts to showing that for all x in the domain of E (and thus in the domain of f): E(-x)=E(x) holds. Proof:
And note: we have done nothing fancier here than first substituting -x for x in the definition of E(x), and then applying very elementary algebra, until we ended up with E(x) again.
Similarly, to show that O(x) is odd, we need to show that for all x in its domain: O(-x)=-O(x). Proof:
If you now take two decompositions of f into even and odd functions, such as and , then it follows that for all x in the domain of f
and thus .
But the difference of two even functions is even, and the difference of two odd functions is odd. Thus and must be both even and odd functions at the same time, from which it follows that and for all x. Hence and ; which is to say that the decomposition of f(x) into an even and an odd part like this is unique.
Maybe this helps - or maybe not...