Term for functions with results between 0 and 1?

Question

Is there a term for functions $f: x \in \mathbb{R}_{\geq 0} \to [0, 1] $ that have the following properties:

1. Strictly monotony
2. $f(0) = 0$
3. $\lim_{x \to \infty} f(x) = 1$

Functions (found here) I think about are:

• $f(x)=1−1/exp(−x)$
• $f(x)=1−1/(1+x)$

The reason I'm asking is that I want to search for such functions but don't no what to search for. Therefore as a starting point it would help to know how functions with results between 0 and 1 are called.

Edit: as requested in the comments here my motivation why I could need functions with such properties. Consider I want to transform the repuation of Stackexchange users into a number between 0 and 1 (a "normalization" which allows me to compare this number) to some other number between 0 and 1 (eg the normalized number of badges). So 0 reputation should be 0; 10 reputation should be 0.1; 1000 reputation should be 0.5, 50K reputation should be 0.9; 1 million repuation should be 0.99, and so forth, but always < 1. If I used $f = log$ this would not bound the values to 1.

Answer 1

I don't think there is any accepted term for these functions, but you could use your own suggestion, after a careful warning. Something like:

For the purpose of this paper, let us introduce a convenient definition. Let us call normalization function a function $f: x \in \mathbb{R}_{\geq 0} \to [0, 1] $ satisfying the following properties:

1. $f$ is strictly monotone,
2. $f(0) = 0$,
3. $\lim_{x \to \infty} f(x) = 1$.