Looking for a function R->[0,1) with particular properties I'm looking for a function with a set of properties and I don't know what to use. Anyone have any clues? It needs to output a probability, range [0,1) (note: strictly not equal to 1 at the upper limit), domain [0,infinity). It needs to satisfy the INADA conditions: f'(x)>0, f''(x)<0, f'(x)->infinity as x->0, f'(x)->0 as x->infinity.
The usual quartile functions (probit, logit) are sigmoid (S-shaped) and so don't satisfy the INADA conditions as they are convex and concave in different domains. Continuously differentiable everywhere the function is defined would be amazing. I feel like I'm missing something obvious, but any help appreciated :)
All the best,
Yuan

How about $x\mapsto \sqrt{\frac{x}{1+x}}$?
