Undergrad Looking for S-shaped function with range 0 to 1 (but not asymptotic)

[Swamp Thing]

At 02:08, this video shows a function that grows from exactly 0 at input x = 0+, up to 1 at $x=\infty$.
Its value and all its derivatives approach 0 as x -> 0. The function is Exp(-1 / x^2).

www.youtube.com/watch?v=Wwg_15a0DJo&t=146s

Q. : What function would have its value and all derivatives = 0 at 0+, then grow with x and attain a value of unity at x -> 1- with all derivatives also tending to zero as x -> 1- ?

 

[mfb]

Take the integral of any smooth function with a compact support. Scale it as needed.

https://en.wikipedia.org/wiki/Bump_function

 

[Swamp Thing]

Thanks, that's some good info.

They give the example of the bump function Exp( - 1 / (1-x^2)).
What is the integral of that? I'm not good at integrating expressions, so I tried Mathematica, which also couldn't come up with a closed form.

It would be nice to have a closed form for the final S-shaped function.

 

[mfb]

There is no closed form of that, but you have a closed form for all derivatives.

You can try to splice together two functions of the shape of Exp(-1 / x^2) and 1-Exp(-1 / x^2), but I'm not sure if you can make all derivatives stay defined.