How to prove that a function is sigmoidal 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.
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 $y(0) = 1/2$.