example of Lipschitz condition
We want to show that for some real constant , and for all ,
Let . Clearly if , the above inequality holds, so assume . Since and are interchangable in the above equation, it can be assumed without loss of generality that .
Additionally, is the Lipschitz constant of .
Assume , since if , it is possible to consider instead of . This also implies that . Let be sufficiently small that and that higher powers of can be ignored. Now,
By the assumption above, . Thus, since and by the definition of the Lipschitz condition,
However, the result from the previous proof gives
Combining these inequalities provides
and the result follows by trichotomy. ∎
|Title||example of Lipschitz condition|
|Date of creation||2013-03-22 17:14:16|
|Last modified on||2013-03-22 17:14:16|
|Last modified by||me_and (17092)|