SIMON-MARAIS 2025 A4

Let $V$ be a finite-dimensional subspace of the real vector space of $C^\infty$ functions from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for all functions $g(x) \in V$, we have $g(2x) \in V$.

Prove that every function in $V$ is a polynomial.

(A function from $\mathbb{R}$ to $\mathbb{R}$ is $C^\infty$ if its $k$-th derivative exists at every point, for all positive integers $k$.)