1. ## create function

can someone help me please build 2 bijection(injective and surjective) functions:
1. from (0, 1) to (0,infinite )
2. from (0,1) to [0,infinite)

thanks.

2. Originally Posted by tukilala
2 bijection(injective and surjective) functions:
1. from (0, 1) to (0,infinite )
2. from (0,1) to [0,infinite)
1. $\displaystyle f(x)=\tan \left( \frac{\pi x}{2} \right)$

3. Originally Posted by tukilala
2. from (0,1) to [0,infinite)
A bijection $\displaystyle g0,\,1)\to[0,\,1)$ is defined by

$\displaystyle g(x)\ =\ \left\{\begin{array}{cl}0 & x=\dfrac12\\[7mm] \dfrac1{n-1} & x=\dfrac1n,\ n\in\mathbb Z,\ n\ge3\\[7mm] x & \mbox{otherwise} \end{array}\right.$

Define $\displaystyle F:[0,\,1)\to[0,\,\infty)$ by $\displaystyle F(0)=0$ and $\displaystyle F(x)=f(x)$ for $\displaystyle 0<x<1$ where $\displaystyle f$ is your bijection in (1). Then the required bijection for (2) is $\displaystyle F\circ g.$