Problem 5 - Basic Analysis: Introduction to Real Analysis

Question

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by $$ f(x):=\left\\{\begin{array}{ll} x \sin (1 / x) & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right. $$ Is \(f\) continuous? Prove your assertion.

Step-by-Step Solution

Answer

Yes, \( f \) is continuous for all \( x \in \mathbb{R} \).

Step 1: Understanding Continuity Definition

A function \( f \) is continuous at a point \( c \) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( |x - c| < \delta \), \( |f(x) - f(c)| < \epsilon \). To determine if \( f \) is continuous at \( x = 0 \), we need to determine whether \( \lim_{x \to 0} f(x) = f(0) = 0 \).

Step 2: Investigate Limit as x Approaches 0

For \( x eq 0 \), the function \( f(x) = x \sin(1/x) \). To analyze \( \lim_{x \to 0} x \sin(1/x) \), note that \( |\sin(1/x)| \leq 1 \). Therefore, \( |x \sin(1/x)| \leq |x| \).

Step 3: Use Squeeze Theorem

Since \( |x \sin(1/x)| \leq |x| \), apply the squeeze theorem:\[ \lim_{x \to 0} -|x| \leq \lim_{x \to 0} x \sin(1/x) \leq \lim_{x \to 0} |x| \]As \( x \to 0 \), both \( \lim_{x \to 0} -|x| = 0 \) and \( \lim_{x \to 0} |x| = 0 \). Thus, by the squeeze theorem, \( \lim_{x \to 0} x \sin(1/x) = 0 \).

Step 4: Conclusion on Continuity at x = 0

Since \( \lim_{x \to 0} f(x) = 0 = f(0) \), \( f \) is continuous at \( x = 0 \). Recall that for all other \( x eq 0 \), \( f \) involves continuous components (product of a continuous function \( x \) and \( \sin(\cdot) \)). Therefore, \( f \) is continuous for all \( x \in \mathbb{R} \).

Key Concepts

Squeeze Theorem

The Squeeze Theorem is a powerful tool in calculus, particularly useful for evaluating limits. It is especially helpful in cases where direct substitution into a function doesn't easily reveal the limit.
Imagine you have three functions: \( g(x) \leq f(x) \leq h(x) \). If you know that \( \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L \), then the Squeeze Theorem tells us that \( \lim_{x \to a} f(x) = L \) as well.
In our problem, we used:

• \( g(x) = -|x| \)
• \( f(x) = x \sin(1/x) \)
• \( h(x) = |x| \)

Applying the Squeeze Theorem, since both \(-|x|\) and \(|x|\) approach zero as \( x \) approaches zero, \( f(x) \) must also approach zero.
This approach enables us to conclude that \( \lim_{x \to 0} x \sin(1/x) = 0 \), providing a crucial step in proving the function’s continuity at this point.

Limits

Limits are the foundation of calculus and critical in understanding continuity. The idea of a limit is all about determining what value a function approaches as the input approaches some point.
When exploring whether a function is continuous, as in our exercise, limits reveal how the function behaves near a specific point — without necessarily reaching that point.
The limit \( \lim_{x \to 0} x \sin(1/x) = 0 \) is key. Here, the limit tells us that regardless of the oscillations caused by \( \sin(1/x) \), the function's magnitude becomes negligibly small as \( x \) approaches zero.
Understanding limits requires:

• Identifying when a limit exists — typically, the left-hand and right-hand limits must match.
• Using logical arguments, like the Squeeze Theorem, when direct computation isn’t straightforward.

The mastery of limits underpins all work in calculus and ensures that calculated values are accurate and meaningful.

Real Analysis

Real Analysis is the branch of mathematics dealing with real numbers and their properties. It lays the groundwork for advanced calculus concepts, such as continuity and limits.
In the context of this exercise, Real Analysis provides the rigorous framework needed to analyze functions deeply. For such a function as \( f(x) = x \sin(1/x) \), we employ concepts from real analysis to definitively determine continuity at \( x = 0 \).
Key concepts in real analysis relevant here include:

• Understanding precise definitions, such as the definition of continuity requiring the limit of a function at a point to equal the function's value at that point.
• Analyzing functions by deconstructing them into simpler or bounding functions, as we did using the Squeeze Theorem.
• Involving a multitude of algebraic and analytical techniques to study convergence and divergence.

A strong grasp of real analysis allows us to confidently apply theorems and computational methods to prove statements like the continuity of the function across its domain.