Bounded and Unbounded Functions

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Lesson Summary: 
We discuss what it means for a function's range to be bounded or unbounded.
Lesson Inputs: 
Functions
Upper and Lower Bounds
Lesson Outputs: 
Properties of Continuous Functions
Sequences
The Logical Framework For Definite Integrals

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Lesson: 

Recall in our lesson on upper and lower bounds that a set of real numbers, $S$, is bounded when for all <Sign in to see all the formulas>, <Sign in to see all the formulas>. Another way of saying this is that for all $s_i$ in $S$, <Sign in to see all the formulas>. Since there is an infinite number of upper and lower bounds, there is an infinite number of values $B$ for which <Sign in to see all the formulas>.

Recall that the range of a function is a set of real numbers. Therefore, a function is said to be bounded in a domain $D$ if the range of the function is bounded. More formally we say that the function is bounded by $B$ if for all $x$ in $D$, <Sign in to see all the formulas>.

Finally, we say that a function is unbounded if for every real number <Sign in to see all the formulas> there is at least one $x$ in the domain of the function for which <Sign in to see all the formulas>.