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I am a tutor for complex analysis this semester and my students had to show as an exercise that every holomorphic and bijective function $f: \mathbb{C} \to \mathbb{C}$ is affine linear, i.e. there exist some $a, b \in \mathbb{C}, a \neq 0$ such that $f(z) = az+ b$.

I really like this exercise and the proof I know uses some fundamental properties of $\mathbb{C}$, i.e. the fundamental theorem of Algebra, Laurent expansion and roots of unity. So to make it clear to my students how wrong this statement is in the reals, I want to give them as many counterexamples of smooth ($C^{\infty}$) bijective functions $\mathbb{R} \to \mathbb{R}$ as possible.

One can for example take $x^n$ for $n \in \mathbb{N}$ odd and sums of this, i.e. polynomials with only odd exponents.

What I am interested in now however is if we can construct examples that are not polynomials. If we allow the function to be less smooth (e.g. $C^1$), one can construct examples such as $\frac{x|x|}{2}$ but I can not find examples that are $C^{\infty}$ so far.

So my question is: do you know any bijective and $C^{\infty}$ function $\mathbb{R} \to \mathbb{R}$ that is not a polynomial?

Edit: Thank you for all your answers, I will present these to my students. I was thinking way too complicated for some reason.

kade
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$\tan(x)$ is smooth bijection between $(-\pi/2,\pi/2)$ to $\Bbb R$. Now choose any smooth bijection $g$ from $\Bbb R$ to $(-\pi/2,\pi/2)$. The compostion $\tan \circ g$ is a smooth bijection from $\Bbb R$ to $\Bbb R$.

General Grievous
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