Knowlpedia

Separates points

A property of a family of functions distinguishing any two different points.

A family F\mathcal F of functions on a set XX separates points if, for every distinct x,y∈Xx,y\in X, there exists f∈Ff\in\mathcal F such that f(x)≠f(y)f(x)\ne f(y).

This condition appears in the for subalgebras of continuous functions.

Examples
  • Real polynomials restricted to [a,b][a,b] separate points because p(t)=tp(t)=t does.
  • Constant functions do not separate two distinct points.