Definition: Let the domain of function be , and  the number set . If there is a positive number , so that for all , there is always

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Then the function is said to be bounded on , or is a bounded function on . Every positive number with the above properties is the bound of the function.

　　Note: According to the inverse proposition of the above definition , if you want to prove that the function is unbounded, you need to prove that for any positive number , exists , so that

.

        If the positive number M with the above properties does  exist, then  is said to be unbounded on , or  is an unbounded function on .

　　If there exists a constant such that for all there is always

( or ) ,

Then the function is said to have an upper bound(or lower bound ) on .

　　It is easy to know that the necessary and sufficient condition for a function  to be bounded on  is that the function  has both an upper bound and a lower bound on .

　　For example, the sine function is bounded within , because for any real number , there is always

;

　　The power function has a lower bound on the interval and no upper bound , so it is an unbounded function .