I need to find a function that always goes through $(0, 0)$ and $(1, 1)$ points. Also it must have a parameter "a" that control the slope. As parameters goes from its first boundary to the second value the slope changes from liner to "step". Please see the picture as it should be.
Functions may go out of $(0, 0)$ $(1, 1)$ area. Hoever the behaviour inside must be as I described. Parameter boundary values can be 0 and 1 or any other values.
How about
$$y=\frac{1}{2}\tanh\left(\frac{1}{1-a}\tanh^{-1}(2x-1)\right)+\frac{1}{2}$$
edit: my previous answer went from $(0,0)$ to $(2,2)$. Also here is a desmos link to play with
https://www.desmos.com/calculator/rezernidg8