Sometimes it is useful to use an alternative but equivalent version of uniform continuity of a function:
Suppose is a function and is defined by . Then is uniformly continuous iff for any , there is a such that .
Suppose is uniformly continuous. Pick any . Then exists with for all . If , then , or , or . The converse is straightforward. ∎
Remark. Note that we could have picked so the inclusion becomes an equality.
Let be open in and set . Pick any . Then has a uniform neighborhood . By the uniform continuity of , there is an entourage with . ∎
Remark. The converse is not true, even in metric spaces.
|Date of creation||2013-03-22 16:43:15|
|Last modified on||2013-03-22 16:43:15|
|Last modified by||CWoo (3771)|