Can we have a pasting lemma for uniform continuous functions

[PKSharma]

In analysis, the pasting or gluing lemma, is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions. Can we have a similar situation for uniform continuous functions?

 

[PKSharma]

The result hold when X has the property that every ball is path connected.

 

[member 587159]

In analysis, the pasting or gluing lemma, is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions. Can we have a similar situation for uniform continuous functions?

In what context are you working? Real numbers? Connected metric/topological spaces?

For example, gluing two closed intervals keeps things uniform continuous. I guess the finite union of compacts will work too.