Is it possible for a function to be separated into two regions of the plane such that a bijection exists between those two regions. In other words, can we separate a curve described by a function into two distinct parts and equate those parts as being equal in size?

I would like to set up a bijection between a curve contained in the unit square, and a curve greater than $1$ in it's domain and range. These curves must be part of the same function but separated like so.

Is there a bijection between these two parts of the function?

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  • $\begingroup$ Do you mean the graph of the function is separated, as happens for some piecewise functions? if so there are simple examples... $\endgroup$
    – coffeemath
    Feb 21 '19 at 2:33
  • 1
    $\begingroup$ There is a bijection between any two regions of the plane, as long as each region contains some open disk. It needn’t be continuous, and the regions needn’t have the same area. For example, there is a bijection between an open disk and a closed disk, and there is a bijection between a disk of radius $0.00001$ and a half-plane. $\endgroup$
    – MPW
    Feb 21 '19 at 3:03
  • $\begingroup$ yes the graph of the function is separated $\endgroup$
    – geocalc33
    Feb 24 '19 at 6:26

Yes. Take the sine curve, $y=\sin x$. Parts of the curve are above $x$-axis and parts below the $y$-axis satisfy the condition. One can easily set up a bijection between them:

The bijection sends $(x, \sin x)\mapsto \big(x+\pi, \sin (x+\pi)\big)$


Yes, it is possible to do that.

The simplest example of this is that consider the line $y=x$ and divide it into two parts- above and below the x-axis.

There is a bijection, namely $x\mapsto -x$ between the two parts. Therefore, they are equal in size.

Hope it helps


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