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 \( X \).