Prove that the function $ g $ satisfying $g(g(x))=2g(x)-x $ is strictly monotonic.

Solution 1:

A Continuous $1-1$ function $f:\Bbb{R}\to \Bbb{R} $ is strictly monotone.

Hint:(Use Intermediate value theorem)

Strategy:

Assume $f$ is not strictly monotone.

Then, there exists $ x,y,z\in \Bbb{R}$ with $ x<y<z $ such that either:

$f(x)\le f(y)\ge f(z)$

Or

$f(x)\ge f(y) \le f(z) $

Now ignore the equality between all of them otherwise it fails to be one-to-one.

Now use intermediate value theorem to contradict that $f$ is one -to-one.

Solution 2:

Hint: Suppose that $g$ is not strictly monotonic. Then, there exist $x_1 < x_2 < x_3$ such that $g(x_1) < g(x_2)$ but $g(x_2) \geq g(x_3)$. Now, make use of the continuity and injectivity of $g$.