absolutely continuous on versus absolutely continuous on for every
Let . Then for all :
Since is continuous on and differentiable on , the mean value theorem (http://planetmath.org/MeanValueTheorem) can be applied to . Thus, for every with , . This yields , which also holds when . Thus, is Lipschitz on . It follows that is absolutely continuous on .
On the other hand, it can be verified that is not of bounded variation on and thus cannot be absolutely continuous on . ∎
|Title||absolutely continuous on versus absolutely continuous on for every|
|Date of creation||2013-03-22 16:12:19|
|Last modified on||2013-03-22 16:12:19|
|Last modified by||Wkbj79 (1863)|