Real Functions

Definition: Upper Bound

A real number $u\in \mathbb{R}$ is an upper bound of a real-valued function $f:D\to \mathbb{R}$ if

$$f(x)\le u\forall x\in D$$

Definition: Lower Bound

A real number $l\in \mathbb{R}$ is an upper bound of a real-valued function $f:D\to \mathbb{R}$ if

$$l\le f(x)\forall x\in D$$

Definition: Bounded Function

A real-valued function is:

• bounded above if it has at least one upper bound;
• bounded below if it has at least one lower bound;
• bounded if it is bounded above and bounded below.