2
$\begingroup$

Could someone tell me if there exists a continuous and surjective function $f:\mathbb{R} \to \mathbb{R}$ such that $\# \{f^{-1}(\left\{y\right\})\} = 3$ for all $y \in \mathbb{R}$?

1 Answers 1

1

How about the following function?

Consider $f(x) = \sin(x \left(\bmod {3 \pi/2} \right)) - \left \lfloor\dfrac{2x}{3 \pi} \right \rfloor$

enter image description here

The plot was made using mathematica $8$.

  • 0
    This shows it can be done. I wonder if there's a nice choice of $k$ for which $f(x)=cos(x)+kx$ would work. I tried with $k=2/3\pi$ and it looked nearly OK on my small screen graphing calculator, making me think there might be an exact choice of $k$ making it work. +1.2012-11-30