Real Function is Expressible as Sum of Even Function and Odd Function

Theorem
Let $f: \R \to \R$ be a real function which is neither an even function nor an odd function.

Then $f$ may be expressed as the pointwise sum of an even function and an odd function.

Proof
Let:

We note that:

Thus $g$ is an even function.

Then:

Thus $h$ is an odd function.

Then:

Hence the result.