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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 $$X$$.

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