Uniform Continuity of h(x)=x3+1 on [1, ∞)

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In summary, the function h(x)=x3+1 is not uniformly continuous on the set [1,infinity) as it is not possible to choose a delta value that satisfies the definition of uniform continuity for all x and y in the set. This is due to the fact that the difference between the function values, |f(x)-f(y)|, becomes zero when x and y are both equal to 1, leading to a contradiction in the choice of delta.
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toddlinsley79
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Homework Statement


Is h(x)=x3+1 uniformly continuous on the set [1,infinity)?

The Attempt at a Solution



Let [tex]\epsilon[/tex]>0. For each x,y in the set [1,infinity) with |x-y|<[tex]\delta[/tex], we would have |(x3+1)-(y3+1)|=|x3-y3|

Now how can I show that this is less than epsilon?
 
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  • #2
It's really important that the question doesn't say the function is uniformly continuous
 
  • #3
Ok so |f(x)-f(y)| = (x3+1)-(y3+1)=x3-y3
Now I'm confused about choosing my delta. I thought I was supposed to choose my delta as epsilon divided by (x3-y3) evaluated at x=1 and y=1 because the boundary is [1,infinity).
However, this would lead to a contradiction because x3-y3=0 and I must choose a delta that is smaller than x3-y3=0 and larger than 0.
 

1. What is the definition of uniform continuity?

The definition of uniform continuity is a mathematical concept that describes a function's behavior in terms of how small changes in the input result in small changes in the output. Specifically, a function f is uniformly continuous if for any given small positive number ε, there exists a corresponding small positive number δ such that whenever the distance between two points in the domain of f is less than δ, the distance between their corresponding points in the range of f is less than ε.

2. How is uniform continuity different from continuity?

Continuity refers to a function's behavior at a single point, where the limit of the function at that point exists and is equal to the value of the function at that point. Uniform continuity, on the other hand, refers to a function's behavior over an entire interval, where small changes in the input result in small changes in the output. In other words, continuity is a local property, while uniform continuity is a global property.

3. Is the function h(x)=x3+1 uniformly continuous on [1, ∞)?

Yes, the function h(x)=x3+1 is uniformly continuous on the closed interval [1, ∞). This can be shown using the definition of uniform continuity, where for any given ε>0, we can choose δ=ε/3. Then, for any two points x1 and x2 in the interval [1, ∞), if |x1-x2|<δ, we have |(x1)^3+1-(x2)^3+1|<ε, which satisfies the definition of uniform continuity.

4. What is the significance of uniform continuity in real analysis?

Uniform continuity is an important concept in real analysis as it allows us to make precise statements about a function's behavior over an entire interval. It is also a key tool in proving many important theorems, such as the Intermediate Value Theorem and the Extreme Value Theorem. Additionally, uniform continuity is closely related to the concept of differentiability, which is essential in calculus and many other areas of mathematics.

5. Can a function be uniformly continuous but not continuous?

No, a function cannot be uniformly continuous but not continuous. This is because uniform continuity is a stronger condition than continuity, and a function must be continuous in order to be uniformly continuous. If a function is not continuous at a point, it cannot be uniformly continuous on any interval containing that point. However, it is possible for a function to be continuous but not uniformly continuous, as demonstrated by the function f(x)=1/x on the interval (0, 1).

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