In the book "A Singular Mathematical Promenade", there's a Theorem of Kontsevich:
It's impossible to find polynomials $P_1(x)$, $P_2(x)$, $P_3(x)$, $P_4(x)$ satisfying the following two conditions:
$1$. For small $x<0$:
$$P_1(x) < P_2(x) < P_3(x) < P_4(x)$$
$2$. For small $x>0$: $$ P_2(x) < P_4(x) < P_1(x) < P_3(x)$$
However, the book goes on to say that if we relax the condition for $P_i$, allowing them to be just smooth $C^{\infty}$ functions, it is possible to find $P_1(x)$, $P_2(x)$, $P_3(x)$, $P_4(x)$ satisfying the above conditions.
I have tried a lot, but without success.
First I considered $P_1(x)$ to be the zero function so I could reduce the problem to just three functions.
I tried a wide array from exponentials, to trigonometrics, logarithms, but nothing works.
I suspect that "traditional" smooth functions, won't satisfy the property because around the origin it all boils down again to (Taylor) polynomials.
Can anyone here provide an explicit quartet of functions satisfying propeties $1$ and $2$? Can they be functions defined by a single branch? Or must we always define two branches?