# Proving that a function is odd

Assume that there exists a function $f:\mathbb{R}\to\mathbb{R}$ that is bijective and satisfies

for all $x$. Here $f^{-1}$ is the inverse function. Show that $f$ is odd.

This was a brain-teaser given to me by a friend. Two other related questions are:

1. Show that $f$ is discontinuous
2. Give an example of such a function (if indeed one exists).

Edit: As an initial idea, maybe approaching the problem graphically would help? A function and its inverse are reflections of each other about $y=x$ on the $x$$y$ plane. Does this lead to anywhere?

Let $\phi = \frac{1+\sqrt5}{2}$, and let
Then $f$ is a bijection, whose inverse is
Checking each case shows that $f(x)+f^{-1}(x) = x$, as required.