# 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

• Note that "Inada" is a person's name, not an acronym -- so there's no need to shout it. – Henning Makholm Apr 5 '13 at 14:43

How about $x\mapsto \sqrt{\frac{x}{1+x}}$?
• Alternatively, for a more gradual approach to 1, there's $\frac{\sqrt{x}}{1+\sqrt{x}}$. – Glen O Apr 5 '13 at 14:42