I need to find a continuous function which takes every real value exactly 2n+1 times, for any $n \in \mathbb{N} $
Thank you in advance
I need to find a continuous function which takes every real value exactly 2n+1 times, for any $n \in \mathbb{N} $
Thank you in advance
Hint:
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I will show you one way to do the case $n=1$. You can generalize it to all $n$.
Start with a function $\phi:[0,1]\to[0,1]$ whose graph is
$\phi$ takes every $y\in(0,1)$ exactly three times, but $0$ and $1$ only two. Now define $f$ on $[k,k+1]$ as $\phi(x-k)+k$, $k\in\mathbb{Z}$.
Same idea:
$f(x) = \sin(x)+ax$, where $a \approx 0.21723$ is the solution of $\sqrt{1-a^2} - a\pi-a\arccos(a) = 0$, so that one local maximum value equals a subsequent local minimum value.