IMC 2026 · Day 2 · P6

(a) There is no such function. The given identity implies that the function is injective. Since it is differentiable and thus continuous, it is also strictly monotone, meaning that $f^{\prime}(x)$ cannot admit both negative and positive values. The latter clearly contradicts the given identity, meaning that such a function does not exist.

(b) We will show an example for such a function. It is reasonable to look for a function of the form $f(x) = k |x|^{a}$ for some $k > 0$, which is differentiable for $a > 1$ with $f^{\prime}(x) = (k a) \operatorname{sign}(x) |x|^{a-1}$ for any $x \in \mathbb{R}$. Since $f(x) \geq 0$ for each $x \in \mathbb{R}$, we see that $$f^{\prime}(f(x)) = (k a) \left( k |x|^{a} \right)^{a-1} = a k^{a} |x|^{a^{2}-a}$$ for all $x \in \mathbb{R}$, so all we need is to pick $a = \frac{\sqrt{5}+1}{2} > 1$ which solves the quadratic equations $a^{2} - a = 1$

and thus $|x|^{a^{2}-a} = |x|$ for all $x \in \mathbb{R}$, and then $k = \frac{1}{a^{\frac{1}{a}}}$ so that $a k^{a} = 1$.