0
$\begingroup$

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

$\endgroup$
  • 1
    $\begingroup$ 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 ? $\endgroup$ – Delta-u May 22 '18 at 15:37
  • $\begingroup$ This works quite well, thanks! $\endgroup$ – user123 May 22 '18 at 19:03
0
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.