I am looking for a function $f$ having the following characteristics:
- $f$ defined on $[0,1]$
- $f(0)=0$
- $f(1)=1$
$ \forall x \in ]0,1[, x <f(x) < 1$
$f$ differentiable on $]0,1]$
- $f'>0$
- $f'(1)=1$
- $\lim\limits_{x\to0} f'(x)=+\infty$
Finally, I will also need an analytical expression of the inverse function $f^{-1}$.
Do you know such function?
A solution is the function $f:[0,1]\to[0,1]$ defined by $$ f(x)=\tfrac12(1+x)\sqrt{x}, $$ whose inverse function $g$ is defined by $$ g(y)=\left(\sqrt{y^2+\tfrac1{27}}+y\right)^{2/3}+\left(\sqrt{y^2+\tfrac1{27}}-y\right)^{2/3}-\tfrac23 $$