I have this problem; and to solve it, I want to erase the possibility of someone drawing a graph of the sigmoid squashing effect in a given scenario, as shown below.
Say there is this set, call it $J$. There is going to be the use of the term $-$ in here, so heads up.
$J$ is the set of all graphs in the domain $[0,1]$ and range $[n_0,n_1]$ where $n_0 >0$, $n_0 < 1$, $n_1 >0$, $n_1 < 1$ and obviously, $n_1 > n_0$
Now, picture the graph of this:
$$1/(1+1/e^{\tan(\pi(x-1/2))})$$
Create a subset that doesn't contain the following:
$$a(1/(1+1/e^{\tan(a\pi(x-1/2))}))$$ where a is any number in the range $[0,1]$.
Let the following exist. Let $J_{-\sigma_{\pi}}$ equal the subset that does not contain the above, and the subset that does contain the above but not $J_{-\sigma_{\pi}}$ is $J_{\sigma_{\pi}}$
Let also there be a distance $r$ that each point in the graph can be away from, maximum the sigmoidal squish and still be counted as $J_{\sigma_{\pi}}$. $r$ is our bias that helps avoid things too close to the sigmoidal squish.
What is the overall impact of switching someone from being able to graph any possible graph contained in $J$, all graphs inside a specific unit square, to only $J_{-\sigma_{\pi}}$, all graphs inside a specific unit square that don't contain that squashing effect, and what is a neat tidy deviation r that doesn't erase too many graphs? Is there a loophole that allows the squashing effect in other ways on the graph? And what does this all look like in math-speak?