If $f$ is a continuous function and $q \neq 0$ such that $f(f(x)) = p f(x) + q x$ $\forall$ $x \in \mathbb{R}$, what can we say about $f$?

Question

If $f(f(x)) = p f(x) + q x$ for all $x \in \mathbb{R}$ such that $p \in \mathbb{R}, q \in \mathbb{R}\backslash\{0\}$ and $f$ is continuous, what can we say about $f$?

We can clearly say that $f$ is unbounded since the relation $\sup \{f\} \geq p \cdot \sup\{f\} + q\cdot\sup\{x\}$ will be inconsistent if $f$ is bounded.

Now that we know $f$ is unbounded, we need to find out if $f$ is monotonous or not. I need help with this part.

Answer 1

we need to find out if $f$ is monotonous or not.

If $f(a)=f(b)$ then $f(f(a)) = f(f(b))$ so $$ 0 = f(f(b)) - f(f(a)) =p\cdot (f(b)-f(a)) + q\cdot (b-a) = q(b-a)$$ since $q\neq 0$, $a=b$. So $f$ is injective. An injective continuous function is monotone.