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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.

Source

p213