It is well-known that a holomorphic function $f:\mathbb{C}\to \mathbb{C}$ which is injective must be of the form $f(z)=az+b$ for some $a,b\in \mathbb{C}$ with $a\neq 0$. See here for a proof.
My question is whether a similar thing holds true for real-analytic functions. In this situation, an example such as $f(x)=x^3$ shows that being analytic and bijective doesn't imply that the inverse is necessarily analytic. With this in mind, I'll state the question as follows: Suppose there exists a real-analytic $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is a bijection and $f^{-1}$ is also real-analytic. Is it the case that there must exist $a,b\in \mathbb{R}$ such that $a\neq 0$ and $f(x)=ax+b$ ?
My intuition says that there could be something like $\arctan(x)$ which satisfies all the requirements except that the domain is $(-\pi/2,\pi/2)$ rather than all of $\mathbb{R}$.