Let $f : \mathbb{R} \to \mathbb{R}$ be a function with all fibres $(\lbrace{x \in \mathbb{R}| f(x) = c\rbrace} = f^{−1}(c)$, $c \in \mathbb{R})$ either empty or consisting of exactly three points. Find such a function which is continuous.
No clue
Asked
Active
Viewed 274 times
1 Answers
2
Just for fun, an infinitely differentiable function of this kind: $f(x) = ax+\cos x$.
The trick is to find $a$ such that each maximal value matches the second minimum to the right of it. Since $f'(x) = a-\sin x$, the maxima are $\sin^{-1}a+2\pi n$ and the minima are $\pi-\sin^{-1}a+2\pi n$. This leads to equation $$a\,(3\pi -2\sin^{-1}a) =2$$ which cannot be solved explicitly; however it is easy to see that a solution exists. Indeed, the left side is a continuous function of $a$ which is equal to $0$ at $a=0$, and equal to $\pi$ at $a=1$. Therefore, there is $a\in (0,1)$ for which equality holds. Numerically, $a = 0.2228328\ldots$