6

I read that a continuous function can't take each of it's values exactly twice. But I don't understand why e.g. take the function $x^2$ and add one point to it very close to $0$ that also has $0$ value (and shift the rest of the function with the difference between the $x$-coordinate of the added point and $0$), then it takes each value twice. Or is it not continuous in this case?

Also could you give me a function in R that takes each of its values three times?

Nesa
  • 1,255
  • Every function that satisfies this property has an infinite number of discontinuities. Refer to http://www.ams.org/journals/proc/1986-098-02/S0002-9939-1986-0854049-8/S0002-9939-1986-0854049-8.pdf – Christian Ivicevic Dec 09 '15 at 18:41
  • But how does mine have an infinite number of them? It has only one shift in it. – Nesa Dec 09 '15 at 18:45
  • What is this point that's very close to $0$? I think that if you actually try to define the function rigorously, you will see that your approach will not create a precisely $1$-to-$2$ function. – Brian Tung Dec 09 '15 at 18:54
  • Where are you adding the point? What does the function do between the new point and 0? You can't just shift it along like that in a well defined way. – Ori Dec 09 '15 at 18:54
  • @ChristianIvicevic: Doesn't that require compactness of domain or something? Otherwise $f(x) = x-\lfloor x \rfloor$, defined on $[0, 2)$, takes on each value exactly twice. – Brian Tung Dec 09 '15 at 18:56
  • @BrianTung That example is discontinuous. $|x|$ on $\mathbb R - 0$ is cts and 2-1. Further, $f:[0,1]\cup[2,3]\to \mathbb R$ with $f:x\mapsto x$ on $[0,1]$ and $f:x\mapsto x-2$ on $[2,3]$ is 2-1 on a compact, but disconnected domain. – Ori Dec 09 '15 at 19:08
  • @Ori: I know it's discontinuous. It does not have an infinite number of discontinuities. Your second example is well taken, though. Connected and compact, then? – Brian Tung Dec 09 '15 at 19:10
  • It seems I am misunderstanding something in regards to the claim from the paper as all examples that come to mind have a finite amount of discontinuities. – Christian Ivicevic Dec 09 '15 at 19:13

1 Answers1

1

Examples of functions which take each their values three times are

$$f(x)=\cot\left(\frac{3\pi}{1+\exp(-x)}\right)$$

as well as

$$f(x)=2\left\lfloor\frac{x}{3\pi}\right\rfloor-\cos\left(3\pi\left\{\frac{x}{3\pi}\right\}\right).$$

Christian Ivicevic
  • 3,123