F(x) = 1/x on (0,1)

Definition

The function f(x) = 1/x is defined for all x in the interval (0,1) and describes a hyperbola that approaches infinity as x approaches 0 from the right. This function is significant in understanding uniformly continuous functions because it demonstrates how certain continuous functions can behave differently under various conditions. Specifically, it highlights the distinction between continuity and uniform continuity in terms of how changes in x affect changes in f(x).

5 Must Know Facts For Your Next Test

1. The function f(x) = 1/x is continuous on the open interval (0,1) but is not uniformly continuous due to its behavior as x approaches 0.
2. As x gets closer to 0, f(x) increases without bound, demonstrating that uniform continuity cannot be guaranteed in this scenario.
3. Uniform continuity requires that the same δ can be applied for all points in the interval, which fails for f(x) = 1/x as it behaves erratically near 0.
4. The discontinuity at x = 0 indicates that while f(x) remains continuous on (0,1), it cannot be uniformly continuous over this interval.
5. In contrast to f(x) = 1/x on (0,1), functions like f(x) = x are uniformly continuous because they do not exhibit such extreme variations.

Review Questions

• How does the behavior of f(x) = 1/x near 0 demonstrate the difference between continuity and uniform continuity?
  • The function f(x) = 1/x is continuous on (0,1) because it has no breaks or jumps within this interval. However, as x approaches 0, f(x) increases without bound, creating a situation where for any chosen ε, there exists a δ such that not all points in the interval can maintain this distance. This illustrates that even though the function is continuous, it fails to satisfy the conditions required for uniform continuity due to its extreme variation near 0.
• Discuss why f(x) = 1/x on (0,1) serves as an example of a function that is not uniformly continuous and identify the implications for analysis.
  • The function f(x) = 1/x on (0,1) is an example of non-uniform continuity because it cannot guarantee that a single δ will work for all points given any ε. Specifically, near x = 0, small changes in x result in large changes in f(x), making it impossible to keep distances consistent throughout the interval. This has implications for mathematical analysis where uniform continuity is crucial; it indicates that caution must be taken when assuming properties of functions based solely on their continuity.
• Analyze how understanding the characteristics of f(x) = 1/x impacts our overall comprehension of uniformly continuous functions and their applications.
  • Understanding the characteristics of f(x) = 1/x helps clarify why some functions may seem well-behaved but still exhibit non-uniformity. It illustrates critical concepts like asymptotic behavior and how limits affect continuity. Recognizing this distinction is vital in applications such as approximation theory and numerical methods where uniform continuity ensures predictable behaviors over ranges. Thus, the insights gained from examining this specific function deepen our appreciation of uniform continuity's significance across various mathematical contexts.

Related terms

Uniform Continuity: A stronger form of continuity where, for any given small distance, a single distance can be found that works for all points in the function's domain.

Cauchy Criterion: A condition used to determine the convergence of sequences, which states that a sequence is convergent if, for every positive ε, there exists an N such that for all m, n > N, the distance between terms is less than ε.

Limit Behavior: The behavior of a function as it approaches a certain point or value, particularly important for analyzing points of discontinuity or asymptotic behavior.