Can we have a pasting lemma for uniform continuous functions

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SUMMARY

The discussion centers on the pasting or gluing lemma in analysis, specifically regarding uniform continuous functions. It is established that two continuous functions can be combined to form another continuous function, particularly when the space X is path connected. The participants confirm that gluing closed intervals maintains uniform continuity, and suggest that finite unions of compact sets also preserve this property. This indicates a broader applicability of the lemma in uniform continuity contexts.

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  • Understanding of uniform continuity in mathematical analysis
  • Familiarity with topological spaces and their properties
  • Knowledge of piecewise functions and their applications
  • Concept of path connectedness in metric spaces
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  • Research the implications of the pasting lemma in uniform continuity
  • Study the properties of path connected spaces in topology
  • Explore applications of piecewise functions in real analysis
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PKSharma
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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?
 
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The result hold when X has the property that every ball is path connected.
 
PKSharma said:
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.
 

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