===== Intervals =====

==== Definitions ====

The //closed interval// of real numbers with lower bound \(a\) and upper bound \(b\) is defined as the set of all real numbers, \(x\), such that \(a \leq x \leq b\).


Similarly, the //open interval// of real numbers with lower bound \(a\) and upper bound \(b\) is defined as the set of all real numbers, \(x\), such that \(a \lt x \lt b\).


==== Notation ====

The notation \([a , b]\) is often used to denote the closed interval of real numbers with lower bound \(a\) and upper bound \(b\). That is:



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


Simlarly, the notation \((a , b)\) is often used to denote the open interval of real numbers with lower bound \(a\) and upper bound \(b\). That is:


\[
(a,b) = \{x \in \mathbb{R} : a \lt x \lt b \}
\]


It is sometimes necessary to work with intervals that are said to be either half-open or half-closed. In such situations, the following notation is commonly used:


\[
(a,b] = \{x \in \mathbb{R} : a \lt x \leq b \}
\]



\[
[a,b) = \{x \in \mathbb{R} : a \leq x \lt b \}
\]