Question

Prove: If \(f\) is continuous on \([a, \infty)\) and \(f(\infty)\) exists (finite), then \(f\) is uniformly continuous on \([a, \infty)\).

Short answer

Question: Prove that if a function f is continuous in an interval \([a, \infty)\) and it has a finite limit as x approaches infinity, then f is uniformly continuous in that interval. Short Answer: To prove that f is uniformly continuous, we showed that for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all pairs of points x and y in the interval, if \(|x - y| < \delta\), then \(|f(x) - f(y)| < \epsilon\). This was achieved by considering two cases: when both x and y are in \([a, M]\) and when at least one of them is in \((M, \infty)\). Since both cases satisfy the property of uniform continuity, the function f is uniformly continuous on the interval \([a, \infty)\).

Step-by-step solution

Use continuity and limit conditions

Recall that f is continuous on the interval \([a, \infty)\), so given any \(\epsilon_1 > 0\), there exists some \(\delta_1 > 0\) so that for all \(x_1, x_2 \in [a, \infty)\), if \(|x_1 - x_2| < \delta_1\), then \(|f(x_1) - f(x_2)| < \epsilon_1\). Since \(f(\infty)\) exists and is finite, there must exist an \(M > a\) such that for all \(x > M\), we have \(|f(x) - f(\infty)| < \epsilon_1 / 2\).

Identify the cases to analyze

We must now consider two cases to prove uniform continuity: 1. Both x and y are in the interval \([a, M]\). 2. At least one of x or y is in the interval \((M, \infty)\).

Prove uniform continuity for the first case

For the first case, since f is continuous on \([a, M]\), we can find a \(\delta > 0\) such that for all \(x, y \in [a, M]\), if \(|x - y| < \delta\), then \(|f(x) - f(y)| < \epsilon\). Therefore, any pair of points in case 1 will satisfy the property of uniform continuity.

Prove uniform continuity for the second case

For the second case, let x and y be in the interval \([a, \infty)\) with at least one of them in \((M, \infty)\). Without loss of generality, assume \(x > M\). Then, we have: \[|f(x) - f(y)| = |f(x) - f(\infty) + f(\infty) - f(y)| \leq |f(x) - f(\infty)| + |f(y) - f(\infty)|\] From the finiteness of \(f(\infty)\), we also have \(|f(x) - f(\infty)| < \epsilon_1 / 2\) and \(|f(y) - f(\infty)| < \epsilon_1 / 2\). Summing these inequalities, we get \(|f(x) - f(y)| < \epsilon_1\). Thus, any pair of points in case 2 will also satisfy the property of uniform continuity.

Conclude the proof

Since we have shown that for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all pairs x,y in both case 1 and case 2, if \(|x - y| < \delta\), then \(|f(x) - f(y)| < \epsilon\). Hence, f is uniformly continuous on the interval \([a, \infty)\). QED.

Key concepts

Continuous Functions

In understanding the landscape of calculus and functions, the concept of a continuous function serves as an essential building block. To call a function continuous, it must pass a specific kind of test: at any point within its domain, the function's value must approach the value at that point. A straightforward way to envision this is to think of drawing the graph of the function without lifting your pen off the paper.

For example, consider a scenario where you are plotting the temperature throughout a day. A continuous function would allow you to graph these variations without any sudden jumps - the reading at any given moment would be a natural follow-up from the previous one. However, if there were any discrepancies, such as a sudden spike or drop that doesn't reflect the actual temperature trend, we would be dealing with discontinuities. In real analysis, continuous functions are extensively studied for their properties and behaviors. Such understanding is vital in many branches of mathematics and applied sciences.

Real Analysis

Exploring the Realm of Real Analysis

Real Analysis is the branch of mathematical analysis dealing with the real numbers and real-valued functions. This discipline lays the foundation for rigorously studying concepts like sequences, series, and continuity. In real analysis, we often talk about 'proving' statements, which means we're deducing them from established mathematical facts using logic.

When tackling problems in real analysis, the aim is to unpack the underlying structure of the real numbers and comprehend the behavior of functions on an infinitely fine scale. From proving basic properties of numbers to explaining the behavior of complex functions, real analysis forms the backbone of advanced mathematical inquiry and provides the toolkit for proving continuity and uniform continuity, as seen in the sample exercise.

Limits and Continuity

Diving into the heart of calculus, the concepts of limits and continuity are intertwined. A limit paints a picture of what a function is heading towards as the input approaches a particular value. It doesn't matter if the function ever actually gets there—what's important is the destination it seems to be aiming for.

Continuity takes this a step further and essentially asks whether the function actually arrives at its anticipated destination. If at every point in its domain the function's limit matches its actual value, the function is continuous. To establish the continuity of a function in practical terms, we often use 'limit proofs', where we show that as we get infinitesimally close to a point, the function's value approaches a certain number—the same number we get if we straightforwardly apply the function at that point.

Epsilon-Delta Definition

The Precision of the Epsilon-Delta Definition

The epsilon-delta definition provides a formal and precise way to define the limits and, by extension, the continuity of functions. This mathematical framework was devised to remove ambiguity and to allow us to discuss continuity with exactitude.

The 'epsilon' represents a very small quantity, a measure of how closely we want to match the function’s output, and the 'delta' signifies a very small interval around the input value. The definition states that for every epsilon, there exists a delta such that whenever the input values are within delta of a point, their corresponding output values are within epsilon of the function's value at that point. This is a cornerstone concept of real analysis and is crucial for rigorously proving statements about limits and continuity, as demonstrated in the sample exercise we discussed.