What is a saturated function? I couldn't find a definition online.
I know that the sigmoid function is saturated but what does it mean.
 A: Saturation is a word used to describe the sharpness of a curve. To be saturated is to be "flattened." While this may sound non-equatable, you are able to solve for the saturation of a curve numerically.
Thus, a saturated function is a function that has been mathematically "smoothed."
Edit: The sigmoid function is called "saturated" because as x approaches infinity it flattens.
A: COMMENT.- The word "saturated", as far as I know, only appears in the context of the quotient topology of a topological space provided with an equivalence relation $\mathcal R$. The "saturated" of  $A\subset  X $ is equal to $p^{-1}(p(A))$ where $p$ is the canonical projection, i.e. all the points of $X$ which are related by $\mathcal  R$ to a point  of $A$.
On the other hand it seems that your question relieved of functions in economics (namely the so called “satisfaction functions”; however it is not impossible that this be known with another word, not literal English translation of the french "saturé").  Here the copy of a terminal examination classes in France (secondary economy, before college):
“Dans un cadre économique, on appelle fonction de satisfaction une fonction $f$ définie et dérivable sur une partie de $\mathbb R$ et à valeurs dans l'intervalle $[0;100]$.
On dit qu'il y a « saturation » lorsque la fonction de satisfaction prend la valeur 100”.
