Math and science::Analysis::Tao::09. Continuous functions on R
Limit points and isolated points (of sets of reals), definition
Limit points
Let \( X \) be a subset of the real line. We say that \( x \) is a limit point (or cluster point) of \( X \) if it is an adherent point of \( X\setminus \{x\} \).
Not to be confused with limit points of sequences.
Isolated points
Let \( X \) be a subset of the real line. We say that \( x \) is an isolated point of \( X \) if \( x \in X \) and there exists some real \( \varepsilon > 0 \) such that \( |x - y| > \varepsilon \) for all \( y \in X\setminus \{x\} \).
Example
3 is an adherent point of the set \( X = (1, 2) \cup \{3\} \), but it is not a limit point of \( X \), since it is not adherent to \( X \setminus \{3\} \). Instead, 3 is an isolated point of \( X \). In comparison, 2 is a limit point, but not an isolated point of \( X \).