## 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:

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

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