Monotonic increasing smooth function such that $f(0) = 0$ and $\lim_{x \to \infty} f(x) = C$ with $C < \infty$

Question

I'm looking for examples of smooth functions $f \colon \mathbb R^+ \to \mathbb R^+$ such that

• $f(0) = 0$
• $\lim_{x \to \infty} f(x) = C$ with $0 < C < \infty$
• $f$ is monotonic increasing

A clear example is $$ f(x) = C\left(1-\exp(-(x / \lambda)^\beta)\right) $$ The CDF of a Weibull random variable with scale parameter $\lambda > 0$ and shape parameter $\beta > 0$.

What are other examples?

Answer 1

Some examples:

1. $\frac{2C}\pi \arctan ax$ for any $a>0$.
2. $C\tanh ax$ for any $a>0$.
3. $\frac{Cx}{1+x}$
4. $1-\exp(-x)$, and appropriate scaled versions.

Answer 2

Maybe by taking $g(x)=C\cdot f(x)$ with $f(x)$ a smooth transition function you will achieve what you want to do, as example: $$f(x)=\begin{cases} 0,\quad x\leq 0, \\ 1,\quad x\geq 1, \\ \dfrac{1}{1+\exp\left(\frac{2x-1}{x^2-x}\right)},\ 0<x< 1 \end{cases}$$

these kind of functions are smooth $f(x)\in C^\infty(\mathbb{R})$ and they achieve a constant value in a finite interval, having a constant value outside, which makes them non-analytic smooth functions.