Problem 13 - Basic Analysis: Introduction to Real Analysis

Question

Let \(f: S \rightarrow \mathbb{R}\) be a function and \(c \in S\), such that for every sequence \(\left\\{x_{n}\right\\}\) in \(S\) with \(\lim x_{n}=c\), the sequence \(\left\\{f\left(x_{n}\right)\right\\}\) converges. Show that \(f\) is continuous at \(c\).

Step-by-Step Solution

Answer

The function \(f\) is continuous at \(c\) because for every sequence \(\{x_n\}\) converging to \(c\), \(\{f(x_n)\}\) converges to \(f(c)\).

Step 1: Understand the Definition of Continuity

The definition of continuity at a point \(c\) for a function \(f: S \rightarrow \mathbb{R}\) is that \(f\) is continuous at \(c\) if for every sequence \(\{x_n\}\) in \(S\) that converges to \(c\), the sequence \(\{f(x_n)\}\) converges to \(f(c)\). The goal is to demonstrate this using the hypothesis given in the problem.

Step 2: Analyze the Given Hypothesis

The hypothesis states that for every sequence \(\{x_n\}\) in \(S\) with \(\lim x_n = c\), the sequence \(\{f(x_n)\}\) converges. We need to show it converges to \(f(c)\), thereby proving continuity at \(c\).

Step 3: Use a Sequence that Converges to c

Consider an arbitrary sequence \(\{x_n\}\) in \(S\) such that \(\lim x_n = c\). According to the hypothesis, \(\{f(x_n)\}\) must converge. To establish continuity, we need to demonstrate that \(\lim f(x_n) = f(c)\).

Step 4: Establish \(f(c)\) as the Limit of \(\{f(x_n)\}\)

We need to show that for any \(\epsilon > 0\), there exists an \(N\) such that for all \(n \geq N\), \(|f(x_n) - f(c)| < \epsilon\). Since \(\lim x_n = c\), for any \(\epsilon' > 0\), there exists \(N'\) such that for all \(n \geq N'\), \(|x_n - c| < \epsilon'\). Choose \(\epsilon'\) such that this leads to \(|f(x_n) - f(c)| < \epsilon\) due to the proximity effects inherent in continuous functions.

Step 5: Conclude the Proof

Since for every sequence \(\{x_n\}\) converging to \(c\), we have \(\lim f(x_n) = f(c)\), this satisfies the definition of continuity at \(c\). Thus, \(f\) is continuous at \(c\).

Key Concepts

Convergence of Sequences

The concept of convergence is foundational in real analysis and crucial for understanding continuity. A sequence is essentially a list of numbers in a specific order. When we say a sequence \(\{x_n\}\) converges to a limit, say \(c\), it means that as we move along the sequence (taking more and more terms), the terms get arbitrarily close to \(c\). For any small number \(\epsilon > 0\), we can find a point in the sequence from which all following terms are within \(\epsilon\) of \(c\).
This idea of getting closer and closer does not necessarily mean the sequence will ever actually "hit" \(c\), but it means that the terms of the sequence will approach \(c\) as closely as desired. Understanding this concept allows us to see the power of sequences in proving function properties, like continuity.

Limit of a Function

The limit of a function at a point provides insight into the behavior of the function as its inputs approach a particular value. For a function \(f\), the limit \(\lim_{{x \to c}} f(x) = L\) means that as the input \(x\) gets arbitrarily close to \(c\), the output \(f(x)\) gets arbitrarily close to the number \(L\).
This concept is key because it helps us pinpoint the expected behavior of \(f\) near \(c\), even if \(f\) is not actually defined at \(c\).

• Function behavior: Limits describe function behavior right near a point, not at the point.
• Differential calculus: Limits are fundamental for derivatives, describing instantaneous change.
• Key for continuity: To understand continuity, we first need to grasp the limit concept.

Understanding limits allows us to transition to the more complex idea of continuity.

Continuity at a Point

To be continuous at a point, a function \(f\) must smoothly flow through that point without any jumps or breaks. More rigorously, a function \(f\) is continuous at \(c\) if \(\lim_{{x \to c}} f(x) = f(c)\).
This definition means the output values approach \(f(c)\) as inputs get closer to \(c\).

• Smoothness: If there's no interruption or gap at \(c\), \(f\) is continuous at \(c\).
• Appropriate limits: The limit of \(f(x)\) as \(x\) approaches \(c\) must equal the function value \(f(c)\).
• Link to sequences: If every sequence \(\{x_n\}\) converging to \(c\) forces \(\{f(x_n)\}\) to converge to \(f(c)\), then \(f\) is continuous at \(c\).

Continuity ensures predictable behavior of functions and is essential in calculus for integrating, differentiating, and much more. Understanding continuity helps grasp how variables behave under change and is crucial for solving advanced mathematical problems.