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If $f$ is a positive continuous real-valued function defined on $\mathbb{R}$ such that $f$ has a continuous derivative on $\mathbb{R}\setminus\{0\}$ and $\int_{-\infty}^{\infty}f(x)dx < \infty$, and if $g(x) = \int_{-\infty}^xf(t)dt$, then what are necessary and sufficient conditions on $f$ that $g$ be a sigmoid?

In particular, is it sufficient for $f’$ to be increasing on the set of negative numbers, and decreasing on the set of positive numbers?

Also: is the situation pretty much the same if we restrict the domain of $f$ to $[-1,1]$?

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Sigmoid is not in general a technical term with a precise definition. It just means S-shaped. –  Rahul Jan 14 '12 at 20:21

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