Uniform Continuity proof, does it look reasonable?

In summary, the proof shows that for each e>0, there exists a d>0 such that |x/(x+1) - a/(a+1)| < e whenever |x-a| < d, proving that the function f(x) = x/(x+1) is uniformly continuous on [0, infinity).
  • #1
spenghali
14
0

Homework Statement


Note: I will use 'e' to denote epsilon and 'd' to denote delta.

Using only the e-d definition of continuity, prove that the function f(x) = x/(x+1) is uniformly continuous on [0, infinity).

Homework Equations





The Attempt at a Solution



Proof:

Must show that for each e>0 there is d>0 s.t.

|x/(x+1) - a/(a+1)| < e whenever x,a are elements of [0, infinity) |x-a| < d.

|x/(x+1) - a/(a+1)| = |(-x+a)/[(x+1)(a+1)]| [tex]\leq[/tex] |-x+a| = |x-a|.

Thus, given e>0, if we choose d=e then,

|x/(x+1) - a/(a+1)| < e whenever |x-a| < d.

This implies that f(x) = x/(x+1) is uniformly continuous on [0,infinity). QED
 
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  • #2
Sure. That works. You could clean up few details, like x/(x+1) - a/(a+1)=(x-a)/((x+1)(a+1)), not (-a+x)/((x+1)(a+1)) and you could also explicitly justify why |(x-a)/((x+1)(a+1))|<=|x-a| but the proof works fine.
 
  • #3
Cool, thanks for the input.
 

1. What is uniform continuity?

Uniform continuity is a mathematical concept that describes the behavior of a function. It means that for any given value of x, the difference between the function's output at that point and its output at any other point within a certain distance from x remains constant.

2. How is uniform continuity different from regular continuity?

Regular continuity only requires that the function's output at x and at points close to x are close together. Uniform continuity, on the other hand, requires that the function's output at x and at points close to x are close together for all x, not just a specific value.

3. Why is uniform continuity important?

Uniform continuity is important because it ensures that a function behaves consistently and predictably. This is particularly useful in mathematical proofs and calculations, as it allows us to make accurate predictions and draw conclusions about the behavior of a function.

4. How do we prove uniform continuity?

To prove uniform continuity, we typically use the epsilon-delta definition. This involves showing that for any given value of epsilon (a small number), there exists a corresponding value of delta (another small number) such that if the distance between two points on the function is less than delta, the difference between their outputs is less than epsilon.

5. Does a uniform continuity proof always look reasonable?

Whether a uniform continuity proof looks reasonable or not depends on the specific function being analyzed and the techniques used in the proof. However, if the proof follows the standard epsilon-delta approach and all the steps are logically sound, then it can be considered reasonable.

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