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

Intervals (of the reals), definition

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

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

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

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

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

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

(a, b) := {x \in 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

09.03.2020