Function graph - meet an even number of times each horizontal line

Question

I got a really interesting as homework.

Find a formula (try to find as shorter formula as you can) for a function $f(x)$, such that its graph meet each line parallel to the $x$ axis (the $x$ axis also counts) an evn number of times!

It is also difficult to find such a graph! Please help!

Note: I already posted this prblem with an other one, but for the other one I got an aswer, but not for this one, so I post it as a seperated question.

Are there any other restrictions? Does the domain have to be $\mathbb{R},$ or can there be gaps? Is the function required to be continuous? — Reinhard Meier, Apr 16 '18 at 21:04
Possible duplicate of Can a continuous function from the reals to the reals assume each value an even number of times? — B. Mehta, Apr 17 '18 at 00:37

score 1 · Answer 1 · answered Apr 16 '18 at 21:13

1

My proposal: $f(x)=x-x^{-1}$ on $\mathbb{R}\setminus\{0\}$

answered Apr 16 '18 at 21:13

Reinhard Meier

8,302

score 0 · Answer 2 · answered Apr 16 '18 at 21:18

0

How about $f(0)=1, f(1)=1$ on $\{0,1\} \to \Bbb R$ There is no continuous function from all of $\Bbb R$ onto all of $\Bbb R$ that takes each value twice. That has been asked here.

answered Apr 16 '18 at 21:18

Ross Millikan

383,099

Reinhard Meier · Answer 3 · 2018-04-16T21:33:59.797

0

Here is another one: $f(x)=2x-\lfloor x\rfloor$ on $\mathbb{R}\rightarrow\mathbb{R}$

If the number of intersections with the parallel lines is given (e.g. $n$), you can use $$ f(x) = nx-(n-1)\lfloor x \rfloor $$

edited Apr 16 '18 at 21:33

answered Apr 16 '18 at 21:22

Reinhard Meier

8,302

score 0 · Answer 4 · answered Apr 17 '18 at 00:34

0

Another option is $f(x) = \log(x^2)$ on $\mathbb{R} \setminus \{0\}$.

answered Apr 17 '18 at 00:34

B. Mehta

13,032

Function graph - meet an even number of times each horizontal line

4 Answers4