Bijection between two parts of a function?

Question

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?

Answer 1

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 $(x, \sin x)\mapsto \big(x+\pi, \sin (x+\pi)\big)$

Answer 2

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