Editing Sigmoid function (section)

== Properties ==
In general, a sigmoid function is [[monotonic function|monotonic]], and has a first [[derivative]] which is [[bell shaped function|bell shaped]]. Conversely, the [[integral]] of any continuous, non-negative, bell-shaped function (with one local maximum and no local minimum, unless [[Degenerate distribution|degenerate]]) will be sigmoidal. Thus the [[cumulative distribution function]]s for many common [[probability distribution]]s are sigmoidal. One such example is the [[error function]], which is related to the cumulative distribution function of a [[normal distribution]]; another is the [[arctan]] function, which is related to the cumulative distribution function of a [[Cauchy distribution]].

A sigmoid function is constrained by a pair of [[horizontal asymptote]]s as <math>x \rightarrow \pm \infty</math>.

A sigmoid function is [[convex function|convex]] for values less than a particular point, and it is [[concave function|concave]] for values greater than that point: in many of the examples here, that point is 0.