# Bijection problem

1. Apr 11, 2005

### gravenewworld

Hi, I'm trying to map all the reals into the interval [0,1]. I figured out that you can map all the numbers in the open interval (0,1) to all the reals by the function tan(pi(x-.5)) (so if I wanted a function from all the reals to (0,1) I could just take the inverse). But this problem is much harder when you consider the closed interval. Is there a way to modify the tangent function I gave to make another bijective function that hits the end points 0 and 1? Would I have to create some sort of peicewise function to remedy this problem?

2. Apr 11, 2005

### matt grime

The problem is you're naturally enough trying to thinking of continuous functions, as those are the ones we meet most.

Why not now map (0,1) to [0,1] or vice versa? It's not too hard once you start thinking of functions in the right way. For instance, lets send the numbers 1,1/2,1/3,1/4... to 1/2,1/3,1/4.... How'd you do that? now define the function on [0,1] using that bit, adn sending x to x otherwise. What can you say about that map?