Does anyone know such a monotonically increasing function $f$ defined on $[-1,1]$ with the following properties:
(1) Increase very slowly on $[-1,a]$, but has a rapid increasing rate near the right boundary $[a,1]$, e.g. exponentially increasing rate, where "$a$" is a number closer to $1$, e.g. $0.9, 0.95$;
(2) Satisfy boundary conditions $f(-1)=0, f(1) = 1$.
Using polynomials (not as fast as exponentials), one can think of $$ f(x) = \left(\frac{x+1}{2}\right)^a , \quad a>0 \, , $$ which has the advantage of being smooth and very simple. With an exponential growth, one can think of $$ f(x) = \frac{a^{b(x+1)}-1}{a^{2b}-1}\, , $$ with $a,b$ positive. One could think of many other functions such as rational functions (many choices are possible).