Let $f: \mathbb{R}\to \mathbb{R}$ be a differentiable odd function, i.e., $f(x) + f(-x) = 0$ for all $x\in \mathbb{R}$ and $f^\prime(x)$ exists for all real $x$. Further assume the followings:
- $ \lim_{x \to \infty} f(x) = \infty, \quad \lim_{x\to -\infty}f(x) = -\infty. $
- $f$ has no critical points on $\mathbb{R}$, i.e, $|f^\prime(x)|>0$ for all $x\in \mathbb{R}$.
One choice of such $f$ is $f(x) =x$. Do there exist any other choices of $f$ such that the above conditions hold.
It seems like the claim is true. I was initially trying to show that with first-order Taylor expansion, but it looks like it is getting me nowhere. Any help will be appreciated. Thanks.