TITLE: Solution to $1-f(x) = f(-x)$
QUESTION [17 upvotes]: Can we find $f(x)$ given that $1-f(x) = f(-x)$ for all real $x$?
I start by rearranging to: $f(-x) + f(x) = 1$. I can find an example such as $f(x) = |x|$ that works for some values of $x$, but not all. Is there a method here? Is this possible?

REPLY [2 votes]: We have $f(x)-1/2=-(1-f(x))+1/2=-f(-x)+1/2=-(f(-x)-1/2)$;
hence $f(x)-1/2$ is an odd function.
So, $f(x)=1/2+\phi(x)$ for some odd function $\phi$.
Clearly, any such function satisfies the original equation.