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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 $a \lt x \lt b$.

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:

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

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