Let $A,B\subset\mathbb{R},n\ge2$.
Let $f:A\to B$ (not necessarily continuous) such that $\forall a\in A,f^{-1}(a)$ is a tuple of $n$ elements.
I know that if $f$ in continuous, for $A=B=\mathbb{R}$ and $n=2$, such a function does not exist. Therefore I was wondering :
When does such a function exist ? When it does, can one give an explicit formula for such a function ?
I'm taking $A=B=\Bbb R$ here. Following your intermediate value theorem proof that no continuous function exists for $n=2$, you can prove similarly that none exists whenever $n$ is even. However, you can easily construct continuous functions for any $n$ odd. Although I've always done so graphically, I'm sure that you could find explicit formulas.
You would have to find a continuous function that violates the horizontal line test. However, it won't have an $n$-tuple at the critical point. Consider the inverse of the function $y = 4x^2 - 4x + 1$. This function's pre-image is a 2-tuple except at $0.5$. The same can be said for all pre-images of polynomials with degree $k: 2 \leq k \leq n$.
