What is a continuous mapping of the real line $(-\infty, \infty)$ to the interval $[0, 1]$? I'm trying out logs and exponentials but they don't seem to work?
Mapping the Real Line to the Unit Interval
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0Seems like $sin^{-1}$ composed with absolute value may work. – 2011-10-23
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0Start with sine or cosine and modify it slightly. – 2011-10-23
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0try a tangent function? – 2011-10-23
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0@Gary: You don’t want $\sin^{-1}$: it goes the wrong way. – 2011-10-23
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0@Brian, right. Thanks. – 2011-10-23
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0@Alice: That has the same problem as Gary’s suggestion, and the inverse tangent doesn’t pick up the endpoints. – 2011-10-23
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0Do I understand correctly that you are looking for a surjective function? – 2011-10-23
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1Notice that a bijection is not possible, though. – 2011-10-23
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2There is no continuous bijection from $(-\infty,\infty)$ to $[0,1]$, [see this thread](http://math.stackexchange.com/questions/42308/continuous-bijection-from-0-1-to-0-1/42310#42310) where it is shown that there is no continuous bijection from $(0,1)$ to $[0,1]$ and notice that $x \mapsto \frac{1}{\pi}\arctan{x} + \frac{1}{2}$ maps $(-\infty,\infty)$ bijectively to $(0,1)$ with continuous inverse $x \mapsto \tan{\pi(x-\frac{1}{2})}$. – 2011-10-23
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0$\dfrac1{1+e^x}$? $e^{-e^{-x}}$? (These are both bijections from $(-\infty,\infty)$ to $(0,1)$.) – 2015-05-27
6 Answers
I take it you want a continuous surjection from $\mathbb{R}$ to $[0,1]$. A simple example of such a function is $f(x)= \begin{cases} 0 & x<0 \\ x & 0 \le x \le 1 \\ 1 & x>1\end{cases}$
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6AKA $\frac{|x|-|x-1|+1}{2}$ :-) – 2011-10-23
$\frac{x+\sqrt{x^2+1}}{2\sqrt{x^2+1}}$ and $\frac{\tanh(x)+1}{2}$ both map $\mathbb{R}$ into $[0,1]$ but not onto. As Phira mentions $\frac{\sin(x)+1}{2}$ maps $\mathbb{R}$ onto $[0,1]$ but is not $1{-}1$. However, you won't find a $1{-}1$ and onto function whose inverse is continuous, since $[0,1]$ is compact and $\mathbb{R}$ is not.
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0One can probably replace the hyperbolic tangent in the second example with the whole gamut of "sigmoidal functions": the arctangent, the logistic function, the error function... :) – 2011-10-23
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0@J.M.: and even the first example :-) – 2011-10-23
$\dfrac{\sin x +1}2$ is a continuous surjective function from the reals to this interval.
$\dfrac{1}{1+e^{-x}}$ is a one-to-one monotonically increasing map from $\mathbb R$ onto the open interval $(0,1)$.
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1The question was about the closed interval, not the open one. – 2012-11-04
All you need is a continuous function that maps $\mathbb{R}$ onto a closed interval; once you have that, adjusting the interval is fairly easy. (For instance, you can use a linear function.) The natural starting points are the sine and cosine functions.
As the question is formulated, any function that would map all number of the real line to a fixed number of $[0, 1]$ would do. A Lo's (down voted by Julian) answer is also a correct answer, since mapping to $(0,1)$, you map to $[0,1]$.
But what user18723 probably meant, but didn't specify, is to have more than just a continuous function. Perhaps the intent was to have a bijective function? But in that case, it's not possible. A good discussion on this situation can be found here.
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0I am quite amazed that you know who's the downvoter. – 2015-05-27
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0It's not true that $[0,1] \subset (0,1)$ – 2015-05-27