Simple Sigmoid function that levels off at specific points

Question

I need to construct a simple Sigmoid function that levels off at specific values of x, as in this curve:

What is the most simple Sigmoid function that I can use for this mission?

Answer 1

I would suggest using a function of the form :

$$ f \, : \, t \in \mathbb{R} \longmapsto \, y_{0} + (y_{1}-y_{0})\frac{1}{1+\exp\Big( - v\big( t-\frac{x_{0}+x_{1}}{2} \big) \Big)} $$

where you adjust the parameter $v$ such that $\vert y_{0} - f(x_{0}) \vert \leq \varepsilon$ and $\vert y_{1} - f(x_{1}) \vert \leq \varepsilon$ for a $\varepsilon > 0$ very small.