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The set $ A = (0, 1]$ and $B = (−∞, ∞)$

Any suggestions are appreciated. For $f:A\to B$

I've tried using $\tan(3(x-0.5))$

However I'm not sure and think it is simpler than this. Some input would be great!

I'm not sure this is the right place but it'd be great if someone could recommend exercises that could help understanding this topic or even a link with some examples.

Thomas Andrews
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John MacPherson
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    Look at this for an idea to get from $(0,1]$ to $(0,1)$. Then it is easy to get to $(-\frac{\pi}{2}, \frac{\pi}{2})$ to which you an apply the tangent. – John Douma Oct 10 '21 at 00:33
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    There won’t be a continuous function. A continuous $1-1$ function $(0,1]\to\mathbb R$ must be either increasing or decreasing, and then $f(1)$ must be either a minimum or maximum value in the image of $f,$ so the image can’t be the entire real line. – Thomas Andrews Oct 10 '21 at 00:40
  • Am I right in understanding that there is no bijection here? The function $tan(3(x-0.5))$ works for everypoint except 0.5 where it is asymptotic.

    Maybe we can show a bijection with one point as an exception?

     – John MacPherson Oct 10 '21 at 00:49
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    There is a bijection, just not a continuous one. And why $3?$ It should be $\tan(\pi(x-0.5)).$ – Thomas Andrews Oct 10 '21 at 00:52
  • And the asymptotic is at $x=1,$ not $x=0.5$ – Thomas Andrews Oct 10 '21 at 00:53
  • That makes sense. I am fairly certain I need to use a piecewise function defined as $f(x) = tan(π(x−0.5)) for (0,1) and 0 when x=1

    Also, I used 3 because I was estimating using maple to visualize the function. Thanks for pointing that out

     – John MacPherson Oct 10 '21 at 01:03

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