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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

3 Answers 3

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Hint:

$$\hspace{11cm} /\/\/\/\.../$$ $$\hspace{5.5cm} /\/\/\/\.../$$ $$ /\/\/\/\.../$$

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    Is this a step function ?2012-10-09
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    @user43758: No, within each interval it is a sawtooth (though it could be sine waves or other shapes). A step function takes each value uncountably many times. This is an excellent hint, it leaves some work to do.2012-10-09
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    you can make it smooth if you want. Basically, you construct a W shape or a sin-type function which takes all the values in $(-1,1)$ exactly 2n+1 times, and then move it to left-right and up-down....Be carefull: the local max-mins are attained less times than 2n+1, so you need to allign the maxes with the mins of the next copy....2012-10-09
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    This is great! +12012-10-09
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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

enter image description here

$\phi$ takes every $y\in(0,1)$ exactly three times, but $0$ and $1$ only two. Now define $f$ on $[k,k+1]$ as $\phi(x-k)+k$, $k\in\mathbb{Z}$.

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    well it will be sin(x-k)+k no ?2012-10-09
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    Actually $\phi(x)=\sin^2(3\,\pi\,x/2)$ on $[0,1]$, but you can take any function with a similar graph.2012-10-09
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    So for every $i \in {{0,n-1}}$, we define on [i,i+1] the function $\phi (x)= sin^2(\frac{3 \pi x}{2}-i)+i$2012-10-09
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    Yes. But what is important is not the particular formula, but the idea of the construction.2012-10-09
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    So how do I proceed in the construction of the formula ?2012-10-09
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    It will be a piecewise function. You define it by its values on each interval $[k,k+1]$. May be there is another example with a unique formula. Look for it.2012-10-09
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    I don't understand how to right the construction process2012-10-09
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    If you insist on a formula, here is one: $f(x)=\sin^2(3\,\pi(x-\lfloor x\rfloor)/2)-\lfloor x\rfloor$, where $\lfloor x\rfloor$ is the **floor function** (greatest integer less that $x$.)2012-10-10
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Same idea:

enter image description here

$f(x) = \sin(x)+ax$, 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.