Problem 9 - Basic Analysis: Introduction to Real Analysis

Question

Let \(c_{1}\) be a cluster point of \(A \subset \mathbb{R}\) and \(c_{2}\) be a cluster point of \(B \subset \mathbb{R}\). Suppose \(f: A \rightarrow B\) and \(g: B \rightarrow \mathbb{R}\) are functions such that \(f(x) \rightarrow c_{2}\) as \(x \rightarrow c_{1}\) and \(g(y) \rightarrow L\) as \(y \rightarrow c_{2}\). If \(c_{2} \in B\), also suppose that \(g\left(c_{2}\right)=L\). Let \(h(x):=g(f(x))\) and show \(h(x) \rightarrow\) Las \(x \rightarrow c_{1} .\) Hint: Note that \(f(x)\) could equal \(c_{2}\) for many \(x \in A\), see also Exercise 3.1 .14

Step-by-Step Solution

Answer

\(h(x)\) approaches \(L\) as \(x \rightarrow c_1\) by the composition of limits from given conditions.

Step 1: Understanding the Problem

We are given that \(c_1\) is a cluster point of a set \(A\) and \(c_2\) is a cluster point of set \(B\). The functions \(f: A \rightarrow B\) and \(g: B \rightarrow \mathbb{R}\) have certain properties as described, and \(h(x):=g(f(x))\). We need to show that \(h(x)\) approaches \(L\) as \(x\) approaches \(c_1\).

Step 2: Recalling Definitions

A cluster point of set \(A\) implies for every \(\epsilon > 0\), there exists \(a \in A\) such that \(0 < |a - c_1| < \epsilon\). The function \(f(x) \rightarrow c_2\) as \(x \rightarrow c_1\) signifies that for every \(\epsilon > 0\), there exists a \(\delta_1 > 0\) such that \(|f(x) - c_2| < \epsilon\) whenever \(0 < |x - c_1| < \delta_1\). Similarly, \(g(y) \rightarrow L\) as \(y \rightarrow c_2\) means for every \(\epsilon > 0\), there exists \(\delta_2 > 0\) such that \(|g(y) - L| < \epsilon\) whenever \(0 < |y - c_2| < \delta_2\).

Step 3: Leveraging the Definitions

Given \(f(x) \rightarrow c_2\) as \(x \rightarrow c_1\), for any \(\epsilon > 0\), there exists \(\delta_1 > 0\) such that \(|f(x) - c_2| < \frac{\epsilon}{2}\) for \(0 < |x - c_1| < \delta_1\). Similarly, since \(g(y) \rightarrow L\) as \(y \rightarrow c_2\), there exists \(\delta_2 > 0\) such that \(|g(y) - L| < \frac{\epsilon}{2}\) for \(0 < |y - c_2| < \delta_2\).

Step 4: Applying Composition of Limits

To prove \(h(x) \rightarrow L\) as \(x \rightarrow c_1\), we consider the composition \(h(x) = g(f(x))\). Since \(f(x) \rightarrow c_2\), there exists a \(\delta_1\) such that \(|f(x) - c_2| < \delta_2\) for \(0 < |x - c_1| < \delta_1\). Within this small neighborhood, we have \(|g(f(x)) - L| < \frac{\epsilon}{2}\).

Step 5: Gathering All Conditions

Choose \(\delta = \min(\delta_1, \delta_2)\). Thus, whenever \(0 < |x - c_1| < \delta\), we ensure \(|h(x) - L| < \epsilon\). The composition of functions implies the continuity needed to bridge \(c_2\) in \(g\), thus, proving \(h(x) \rightarrow L\) as \(x \rightarrow c_1\).

Step 6: Conclusion

By showing that \(|h(x) - L| < \epsilon\) for any \(\epsilon > 0\) as \(x \rightarrow c_1\), we confirm that \(h(x)\) approaches \(L\). Hence, \(h(x) \rightarrow L\) as \(x \rightarrow c_1\).

Key Concepts

Cluster Point

In real analysis, understanding cluster points is essential for grasping deeper concepts of convergence and continuity. A cluster point of a set is a point where other points of the set get infinitely close, but not necessarily touch.
Consider a subset \( A \) of the real numbers \( \mathbb{R} \), a point \( c_1 \) is a cluster point of \( A \) if, for every \( \epsilon > 0 \), there exists a point \( a \) in \( A \) such that \( 0 < |a - c_1| < \epsilon \). This means that every neighborhood around \( c_1 \) will always contain another different point from \( A \), no matter how small the neighborhood gets.

• Cluster points are crucial because they indicate where a sequence can potentially converge.
• They play a key role in defining limits, as in the exercise where \( c_1 \) and \( c_2 \) are cluster points leading towards certain limits of functions \( f(x) \) and \( g(y) \).

Functions and Continuity

The concept of continuity is fundamental in the study of functions within real analysis, particularly when dealing with limits. A function is continuous at a point if it doesn’t 'jump' or have breaks at that point. More formally, a function \( f: A \rightarrow B \) is continuous at a point \( c_1 \) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( |x - c_1| < \delta \), it holds that \( |f(x) - f(c_1)| < \epsilon \).

• Continuity ensures that small changes in input lead to small changes in output, hence reflecting a kind of smooth behavior across the domain.
• In the exercise, the continuity of functions \( f \) and \( g \) ensures that the composition \( h(x) = g(f(x)) \) behaves predictably as \( x \) approaches \( c_1 \).

Understanding continuity helps in applying limits effectively, especially when dealing with composed functions as outlined in the exercise.

Composition of Functions

The composition of functions involves taking two functions and creating a new function by applying one function to the result of another. If you have two functions \( f: A \rightarrow B \) and \( g: B \rightarrow C \), their composition \( h(x) = g(f(x)) \) involves applying \( f \) to \( x \), and then applying \( g \) to the result of \( f(x) \).

• Compositions are significant in real analysis as they allow the construction of complex functions from simpler ones.
• The limit behavior of the composite function \( h(x) \) can be deduced by understanding the individual limits of \( f(x) \) and \( g(y) \), particularly using the continuity of these functions as \( x \) approaches \( c_1 \).
• In the provided exercise, this composition allows us to conclude that \( h(x) \rightarrow L \) when \( x \rightarrow c_1 \), leveraging the limits of the individual functions \( f \) and \( g \).

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