Hey I'm looking for an analytical (or at least with an analytical inverse) continuous function that has this "landscape" please, and I welcome even differential equations that this function would respect :
Meaning : $$\lim_{r\rightarrow \pm\infty} f(r)=0$$, $$\lim_{r\rightarrow \pm\infty} f'(r)=0$$ and $f(0)=0$ and $f(\pm a)=b$ with $a>0$ and $b>0$, if we make it simple : $$f(\pm 1)=\pm 1$$.
Thank you in advance
Try something like $$ f(x)=-\frac{x}{1+x^2} $$ Or $$ f(x)=-x\exp(-\lambda x^2),\quad\lambda>0. $$