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

In summary, uniform continuity is a type of continuity in mathematics that requires a function to have a constant rate of change over a given interval. It differs from regular continuity by requiring this constant rate of change to be maintained throughout the entire interval. Lipschitz continuity, on the other hand, is a stronger condition that requires the rate of change to be bounded by a specific constant value over the entire interval. To prove uniform continuity, one must show that for any given value of epsilon (ε), there exists a value of delta (δ) that satisfies the condition of small changes in input value resulting in small changes in output value. Not all continuous functions can be considered uniformly continuous, as there are many continuous functions that do not meet the necessary conditions
  • #1
PsychonautQQ
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Homework Statement


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

Homework Equations

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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bump because i don't want 666 post count
 
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