Lipschitz condition and differentiability result
About lipschitz continuity of differentiable functions the following holds.
Suppose that is lipschitz continuous:
Then given any and any , for all small we have
Hence, passing to the limit it must hold .
On the other hand suppose that is bounded on :
Given any two points and given any consider the function
For it holds
Applying Lagrange mean-value theorem to we know that there exists such that
and since this is true for all we get
which is the desired claim. ∎
|Title||Lipschitz condition and differentiability result|
|Date of creation||2013-03-22 13:32:42|
|Last modified on||2013-03-22 13:32:42|
|Last modified by||paolini (1187)|