# Need a boundary layered function

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 $f(-1)=0, f(1) = 1$.

• Is $C \left(e^\frac{x-1}{\epsilon}-e^\frac{-2}{\epsilon}\right)$ with $0< \epsilon \ll 1$ ok ? There is a lot of such functions, do you need any additional property ? – Delta-u May 22 '18 at 15:37
• This works quite well, thanks! – user123 May 22 '18 at 19:03

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).