Functions of bounded variation

Example of BV Function 

                                                          THEOREMS

Theorem 1: A function f:[a,b]→R is a Function of bounded variation on

                   [a,b],then it is bounded on [a,b]

                   But the converse is not true.

Theorem 2: A function f:[a,b]→R is monotone on [a,b] ,then it is   

                    a  Function of bounded variation on [a,b].

                   converse???

Theorem 3: A function f:[a,b]→R satisfy Lipschitz condition on [a,b]

                   ,then it is a  Function of bounded variation on [a,b].

                   But the converse is not true.

Theorem 4: If f, g are Function of bounded variation on [a,b] ,then f+g ,f-g  

                    ,fg ,cf (c scalar)  are Function of bounded variation on [a,b] .

                    Is f/g also ???

Theorem 5:  A function f :[a,b]→R be continuous on [a,b] ,f '(derivative   

                   of  f) exists on (a,b), then f is a BV-function on [a,b] .

example of bv function
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