continuous derivative implies bounded variation
is of bounded variation,
Let be an arbitrary partition of , with the points
Thus the total variation satisfies
whence is of bounded variation on the interval .
. Define the functions and by setting
We see that these are non-negative and that . Define then the functions and on by
Because and are non-negative, the functions and are monotonically nondecreasing. We have also
whence . Since and are by their definitions continuous, the monotonic functions and have continuous derivatives , . So and fulfil the requirements of the theorem.
|Title||continuous derivative implies bounded variation|
|Date of creation||2013-03-22 17:56:32|
|Last modified on||2013-03-22 17:56:32|
|Last modified by||pahio (2872)|