Bijection between (0,1) and [0,1) in R?

[Colleen G]

Homework Statement

I need to find a bijection between (0,1) and [0,1) in R. It can go in either direction since it is a bijection.

Homework Equations

I can't think of any equations at all!

The Attempt at a Solution

Something like f(x) = 1/[(1/x)+1] for x in A
x for x not in A
where A={1/2, 1/3, 1/4, ...}

Having an issue with zero though.

 

[LCKurtz]

Homework Statement

I need to find a bijection between (0,1) and [0,1) in R. It can go in either direction since it is a bijection.

Homework Equations

I can't think of any equations at all!

The Attempt at a Solution

Something like f(x) = 1/[(1/x)+1] for x in A
x for x not in A
where A={1/2, 1/3, 1/4, ...}

Having an issue with zero though.

Not sure which direction you are trying to do. For $[0,1)\rightarrow (0,1)$ try starting with $0\to 1/2$, $1/2\to 1/3$ etc.

 

[HallsofIvy]

The crucial point here is that the set of rational numbers, between 0 and 1, is countable. Write the rational numbers as "$a_1, a_2, a_3, ...$" and map $a_1$ to 0 and $a_{n+1}$ to $a_n$. Map each irrational number to itself.

 

[SammyS]

Having an issue with zero though.

Simplify $\displaystyle\ \frac{1}{\displaystyle\frac{1}{x}+1}\$ to get x out of the denominator.

Although f(0) is undefined, $\displaystyle\ \lim_{x\to 0}\,f(x)=0\$, so f(x) has a removable discontinuity at x=0 .