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

What is uniform continuity?

Uniform continuity is a type of continuity in mathematics that describes the behavior of a function over a certain interval. It means that the function's values change smoothly and consistently within the given interval.

How is uniform continuity different from regular continuity?

Uniform continuity differs from regular continuity in that it requires the function to have a constant rate of change over the entire interval, rather than just at a single point. This means that small changes in the input value will result in small changes in the output value throughout the interval.

What is the difference between uniform continuity and Lipschitz continuity?

The main difference between uniform continuity and Lipschitz continuity is that uniform continuity only requires the function to have a constant rate of change, while Lipschitz continuity requires the rate of change to be bounded by a specific constant value over the entire interval. This makes Lipschitz continuity a stronger condition than uniform continuity.

How do you prove uniform continuity?

To prove uniform continuity, you must show that for any given value of epsilon (ε), there exists a value of delta (δ) such that for any two points within the given interval, the absolute difference in their corresponding function values is less than or equal to epsilon. This can be done using the definition of uniform continuity or by using the Cauchy criterion.

Can all continuous functions be considered uniformly continuous?

No, not all continuous functions are uniformly continuous. In order for a function to be considered uniformly continuous, it must meet the specific conditions described above. There are many continuous functions that do not meet these conditions, such as functions with a discontinuity or functions with a variable rate of change within the interval.

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