Definition of sigmoid function

Question

Wikipedia gives the following definition for a sigmoid function:

A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a non-negative derivative at each point.[1] A sigmoid "function" and a sigmoid "curve" refer to the same object.

It provides the following properties:

In general, a sigmoid function is monotonic, and has a first derivative which is bell shaped. A sigmoid function is constrained by a pair of horizontal asymptotes as $x\rightarrow \pm \infty$.

The sigmoid function is convex for values less than 0, and it is concave for values more than 0. Because of this, the sigmoid function and its affine compositions can possess multiple optima.

The given definition doesn't seem to be sufficient to provide those properties. For instance, since multiple curves are allowed, the derivative could have multiple peaks and not be bell-shaped.

German Wikipedia says:

Im Allgemeinen ist eine Sigmoidfunktion eine beschränkte und differenzierbare reelle Funktion mit einer durchweg positiven oder durchweg negativen ersten Ableitung und genau einem Wendepunkt.

In general, a sigmoid function is a bounded and differentiable real function with a consistently positive or consistently negative first derivative and exactly one inflection point.

Is this definition sufficient? Is it a common definition?

Also, how common is it to use a general definition like this instead of saying that the sigmoid function is the logistic function $1/(1 + e^{-x})$?

Related: Why is the logistic function a special case of the sigmoid function?

Answer 1

An example of a function matching the first definition by not matching the subsequent description is $$f(x)=\frac{\tanh(x+100)+\tanh(x-100)+2}4.$$

(The red curve is $f(x)$ and the blue curve is $f'(x)$.)

A meta-question is, how reliable are Wikipedia entries? Here we have an instance of incoherence within a single W article; instances of cross-article incoherence are also common. The cause of both is the fact that these articles are written by committee, over a period of time, by people with different interests and areas of competence.

The right way to use Wikipedia mathematics articles is anthropologically: they are evidence that some people or groups of people have certain beliefs or ideas, but are not statements of universal truth. In some countries pea soup is green, in others yellow. Statements of the form "pea soup is yellow" found in an encyclopedia of cookery should be understood in context.

In the present instance it is clear that "sigmoid function" is a vague notion, of uncertain meaning to even the author of the Wikipedia article. If you are writing a paper for publication, you can strive to be more precise, using tricks like "Here we take sigmoid function to mean this and that..."