How to prove that a function is sigmoidal

Question

Given a function, how to prove that it is sigmoidal in nature. So far, my approach has been to verify if the properties of sigmoidal functions hold:

1)That it is monotonic

2)That it is constrained by a pair of horizontal asymptotes

3)That it has a first derivative that is "bell" shaped

4)That it is convex for values less than 0 and concave for values more than 0

5)That a Sigmoid function and its affine compositions can posses multiple optima.

For example this is one of the functions that I am trying to verify:

Trying to prove all the properties seems like time consuming especially in exam scenarios. I suppose there is a better approach that exists and I should follow. Please suggest me what other approach could be used.

Thank You.

Answer 1

I'm not so sure about how close this is to what you want but one way to prove if a function $y = f(x)$ is the sigmoid equation, is to check if it's a solution of the ordinal differential equation:

$$ \frac{dy}{dx} = y \; (1-y) $$

with initial condition $y(0) = 1/2$.