Function as a sum

Question

We know that a (continuous) function $f$ can be written as the sum of a (continuous) odd function and a (continuous) even function, we let $h(x)=\frac {f(x)-f(-x)}{2}$ and $g(x)=\frac {f(x)+f(-x)}{2}$.

Do you know any similar properties of a function, with tricks as such, that are good to know?

Answer 1

Here's one:

You can break a function into positive and negative parts. Take $f^+(x)=f(x)$ if $f(x)>0$ and $f^+=0$ otherwise.

Take $f^-(x)=-f(x)$ if $f(x)<0$ and $f^-=0$ otherwise. Then $f=f^+-f^-$.

This also preserves continuity.