Uniform Continuity of f(x) = 1/(|x|+1) on R: Epsilon-Delta Proof

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SUMMARY

The function f(x) = 1/(|x|+1) is proven to be uniformly continuous on R using an epsilon-delta proof. The proof begins by letting ε > 0 and aims to find a δ such that for all x, y in R, if |x - y| < δ, then |f(x) - f(y)| < ε. The discussion emphasizes the need for algebraic manipulation to identify an appropriate δ that satisfies the uniform continuity condition.

PREREQUISITES
  • Epsilon-delta definition of continuity
  • Understanding of real-valued functions
  • Basic algebraic manipulation skills
  • Knowledge of limits and continuity concepts
NEXT STEPS
  • Study the epsilon-delta definition of uniform continuity in detail
  • Explore examples of uniform continuity proofs for different functions
  • Learn about the implications of uniform continuity in analysis
  • Investigate the relationship between uniform continuity and differentiability
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Students in calculus or real analysis, educators teaching continuity concepts, and mathematicians interested in function properties and proofs.

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Homework Statement


Prove that f(x) = 1/(|x|+1) is uniformly continuous on R.

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The Attempt at a Solution


This needs to be an e-d proof (epsilon-delta).

So I suppose we should start with let e>0, then we want to find a d such that for all x,y in R, if |x-y|<d then |f(x)-f(y)|<e.

I'm having trouble locating a d that will work, is there some algebra trick or other type of trick that can help me?
 
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