Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for all real $x$:

$$
f\left(x^{3}+x\right) \leqslant x \leqslant(f(x))^{3}+f(x)
$$
Let $\mathrm{g}: \mathrm{x} \mapsto x^{3}+x$. The function $g$ is continuous and strictly increasing on $\mathbb{R}$, and $g(\mathbb{R})=\mathbb{R}$. Therefore, $g$ is a bijection from $\mathbb{R}$ to $\mathbb{R}$. We denote $g^{-1}$ as its inverse function. Then, $g^{-1}$ is also strictly increasing on $\mathbb{R}$.

Let $f$ be a potential solution. Then, the statement becomes: for all real $x$, we have $f(g(x)) \leqslant x \leqslant g(f(x))$. By applying the first inequality to $g^{-1}(x)$, we get $f(x) \leqslant g^{-1}(x)$. By composing with $g^{-1}$ (which is increasing) the second inequality, we get $g^{-1}(x) \leqslant f(x)$.

Therefore, $f(x)=g^{-1}(x)$.

Conversely, it is easy to verify that $g^{-1}$ is indeed a solution to the problem.