How can I construct a specific sigmoid function?

Intuitive explanation of proof of Abel's limit theorem Can a function be applied to itself? clarification of algebraic closure and algebraically closed field Abelian category without enough injectives Bounded linear operator commuting with every compact operators Solve $x+3y=4y^3,y+3z=4z^3 ,z+3x=4x^3$ in reals $X,Y$ independent then $X+Y$, $X-Y$ independent as well? Prove that $|e^{i\theta_1}-e^{i\theta_2}| \leq |\theta_1 - \theta_2|$ Sums $\sum_{k = 0}^n k^t {n \choose k}$ where $t$ is a positive integer Question about a basis for a topology vs the topology generated by a basis? Prove that if a sequence $\{a_{n}\}$ converges then $\{\sqrt a_{n}\}$ converges to the square root of the limit. How is $O(\log(\log(n)))$ also $O( \log n)$?