[Math] Every function is the sum of an even function and an odd function in a unique way

Question

It is known that every function $f(x)$ defined on the interval $(-a,a)$ can be represented as the sum of an even function and an odd function. However

How do you prove that this representation is unique?

Thanks for your help.

Answer 1

If $f = g_1 + g_2 = h_1+h_2$ where $g_1,h_1$ are even and $g_2,h_2$ are odd then $$ g_1 - h_1 = g_2 - h_2 \tag{1} $$ where the left-hand side of $(1)$ is even and the right-hand side is odd, hence both sides are just $0$. Indeed, it is easy to show just from the definitions that any function which is both even and odd must be a constant $0$.