How does one prove that the decomposition of a function $f(x)=f_{even}(x)+f_{odd}(x)$ on a sum of even and odd functions is unambiguous? I'm unsure of where to begin.
Pertinent definitions:
$f_{even}(x)=\dfrac{f(x)+f(-x)}{2}$
$f_{odd}(x)=\dfrac{f(x)-f(-x)}{2}$
Best Answer
Suppose that $$f(x)=E(x)+O(x),$$ for all $x$, where $E(x)$ is even and $O(x)$ is odd. Then for all $x$, $$f(-x)=E(-x)+O(-x)=E(x)-O(x).\tag{2}$$
Add. We get $f(x)+f(-x)=2E(x)$, and therefore $E(x)=\frac{f(x)+f(-x)}{2}$.