Continuity On Compact Domains. Infinite Limits. Uniform Cont

Continuity on compact domains. Infinite limits. Uniform continuity Definition 17

Let a function f : I → R. We say that f is continuous on I if f is continuous at every point of I, that is to say, for every x ∈ I, and for every ε > 0, there exists δ > 0 such that for all y ∈ I, |x − y|

We say that f is uniformly continuous on I if for every ε > 0, there exists δ > 0 such that for all x, y ∈ I, |x − y|

Examples demonstrate this distinction. The function f(x) = 1/x is continuous on (0, 1], but not uniformly continuous on this interval. The function f(x) = x² is continuous on R but not uniformly continuous on R. Notably, when the domain is compact, continuous functions are also uniformly continuous, as indicated by Theorem 69, which guarantees that continuity on a compact set implies uniform continuity.

Theorem 69 states: Suppose f : D → R is continuous on D and K ⊆ D is compact. Then f is uniformly continuous on K. The proof uses a contradiction approach assuming f is not uniformly continuous on K, and leverages the compactness of K to extract convergent subsequences, leading to a contradiction with the continuity of f.

Furthermore, on compact sets, continuous functions reach their bounds—suprema and infima. Theorem 71 confirms: If f : D → R is continuous on a compact set K, then f is bounded on K, and there exist points y, z ∈ K such that f(y) = sup{f(x) | x ∈ K} and f(z) = inf{f(x) | x ∈ K}. The proof involves the properties of compactness ensuring the necessary limit points are within K, and the continuity of f guarantees the attainment of extrema.

In the context of limits, infinite limits are considered a special case. A sequence {x_n} approaches infinity if for every M ∈ R, there exists N ∈ N such that for all n ≥ N, x_n ≥ M. Similarly, {x_n} approaches negative infinity if for all M, sufficiently large n satisfy x_n ≤ M. This notion extends to functions: if f(x) tends to infinity as x approaches a point or as x tends to infinity, it is expressed through neighborhoods that eventually contain all sufficiently large values.

Examples include the sequence {n^p} for any p ∈ N diverging to infinity, and functions such as f(x) = √(x² + 1), which tend to infinity as x approaches infinity. These limits play a crucial role in understanding the behavior of functions at boundary points and at infinity, especially when analyzing asymptotic behavior.

Implications of infinite limits in analysis

Understanding infinite limits aids in characterizing the behavior of functions near points where they are unbounded, as well as at infinity. For instance, limits like lim x→∞ 1/x = 0 describe the decay of functions at infinity, providing insight into convergence and stability in various applications.

Additionally, classifying discontinuities based on limits is essential for analyzing functions' behavior. Removable discontinuities occur where the limits from the left and right exist and are finite but do not match the function value at that point; such discontinuities can often be "fixed" by redefining the function at that point.

Jump discontinuities happen when the left and right limits exist but are not equal, such as the Heaviside step function. Discontinuities of the second kind involve cases where these one-sided limits do not exist or are infinite, exemplified by functions like the Dirichlet function or sinusoid near zero.

Conclusion

Continuity, uniform continuity, and limits are fundamental concepts in real analysis that describe how functions behave locally and asymptotically. Compactness provides powerful tools to establish uniform continuity and the existence of extrema. Infinite limits extend the framework to unbounded behaviors, essential for a comprehensive understanding of functions and their applications across mathematics and sciences.

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