Separates points

A family $\mathcal{F}$ of functions on a set $X$ separates points if for every pair of distinct points $x,y\in X$ there exists $f\in\mathcal{F}$ such that $f(x)\ne f(y)$.

This property is often imposed on a subalgebra of continuous functions in approximation theorems, notably the Stone–Weierstrass theorem . Intuitively, separating points means the family contains enough functions to distinguish elements of $X$ by their images.

Examples:

• The set of real polynomials restricted to $[a,b]$ separates points of $[a,b]$ (if $x\ne y$, the polynomial $p(t)=t$ already satisfies $p(x)\ne p(y)$).
• The family of constant functions on $X$ does not separate points (all constants take the same value at every point).