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Math and science::Analysis::Tao::09. Continuous functions on R

# Intervals (of the reals), definition

Let $$a, b \in \mathbb{R}^*$$ be extended real numbers.

We define the closed interval $$[a, b]$$ by

$[a, b] := \{ x \in \mathbb{R}^* : a \leq x \leq b\}$

We define the half-open intervals $$[a, b)$$ and $$(a, b]$$ by

$$[a, b) := \{x \in \mathbb{R}^* : a \leq x \leq b\};$$ and $$(a, b] := \{x \in \mathbb{R}^* : a \le x \leq b \}$$

And we define the open interval $$(a, b)$$ by

$(a, b) := \{ x \in \mathbb{R}^* : a < x < b \}$

We call $$a$$ the left endpoint and $$b$$ the right endpoint.

If $$a$$ and $$b$$ are real numbers (not $$\infty$$ or $$-\infty$$), then all intervals above are subsets of the real line.

The real line itself is the open interval $$(-\infty, \infty)$$, and the extended real line is the closed interval $$[-\infty, \infty]$$.

We sometimes refer to intervals where one endpoint is infinite as being a half-infinite interval, and intervals where both endpoints are infinite as being a double-infinite interval; all other intervals are bounded intervals.

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