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I am looking for a function $f(x) \in [0,1]$ when $x \in (-\infty, +\infty)$.

$f(x)$ increases very fast when $x$ is small starting from $-\infty$, and then very slow and eventually approach $1$ when $x$ is infinity.

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    $\begingroup$ What about the continuity and differentiability of $f(x)$? Any restrictions? $\endgroup$ – SchrodingersCat Dec 24 '15 at 12:52
  • $\begingroup$ Look up sigmoid function $\endgroup$ – Alex Dec 24 '15 at 13:05
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$$f(x) = \dfrac1{1+e^{-x}}$$ is one possible candidate.

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Consider the function $f(x) = \frac{\arctan(x)}{\pi} + \frac{1}{2}$

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The function $$f(x)=\frac{1}{1+a^{-x}}$$ has such a property $\forall \, a>1 , a \in \mathbb{R^+}$.

The function by Leg where $a=e$ is a special case of this type of functions.

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