Can You Create a Bijective Function from the Reals to the Interval [0,1]?

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This discussion centers on the challenge of creating a bijective function from the set of real numbers to the closed interval [0,1]. The user successfully identifies the function tan(π(x - 0.5)) to map the open interval (0,1) to all reals and seeks to modify this approach to include the endpoints 0 and 1. The conversation suggests that a piecewise function may be necessary to achieve this goal, emphasizing the importance of continuous functions in this context. Additionally, alternative mappings from (0,1) to [0,1] are proposed, demonstrating the flexibility of function definitions.

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gravenewworld
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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?
 
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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, let's 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?
 

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