uniformly continuous on is roughly linear
Uniformly continuous functions defined on for are roughly linear. More precisely, if then there exists such that for .
Proof: By continuity we can choose such that implies .
Let , choose to be the smallest positive integer such that . Then
so that we have
Note we can extend this result to if is differentiable at 0.
|Title||uniformly continuous on is roughly linear|
|Date of creation||2013-03-22 15:09:41|
|Last modified on||2013-03-22 15:09:41|
|Last modified by||Mathprof (13753)|