Problem 16 - Basic Analysis: Introduction to Real Analysis

Question

Suppose \(f: S \rightarrow \mathbb{R}\) and \(g:[0, \infty) \rightarrow[0, \infty)\) are functions, \(g\) is continuous at \(0, g(0)=0,\) and whenever \(x\) and \(y\) are in \(S\) we have \(|f(x)-f(y)| \leq g(|x-y|) .\) Prove that \(f\) is uniformly continuous.

Step-by-Step Solution

Answer

The function \(f\) is uniformly continuous because \(g\) being continuous at 0, with \(g(0)=0\), ensures \(|f(x)-f(y)| < \epsilon\) for sufficiently close \(x, y\).

Step 1: Understand the definition of Uniform Continuity

A function \(f\) from a set \(S\) to the real numbers \(\mathbb{R}\) is uniformly continuous if for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all \(x, y \in S\), whenever \(|x - y| < \delta\), then \(|f(x) - f(y)| < \epsilon\).

Step 2: Interpret the given inequality and the function \(g\)

We know from the problem statement that \(|f(x) - f(y)| \leq g(|x-y|)\) for \(x, y \in S\). The function \(g\) is continuous at 0 and \(g(0) = 0\). This implies that as \(|x-y| \to 0\), \(g(|x-y|) \to 0\).

Step 3: Use the continuity of \(g\) at 0 to find \(\delta\)

Since \(g\) is continuous at 0 and \(g(0) = 0\), for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \(0 \leq t < \delta\), \(g(t) < \epsilon\). We will use this \(\delta\) to prove uniform continuity of \(f\).

Step 4: Apply uniform continuity definition to \(f\)

Let \(\epsilon > 0\) be given. Using the continuity of \(g\) at 0, choose \(\delta > 0\) such that \(g(t) < \epsilon\) whenever \(0 \leq t < \delta\). Now, if \(x, y \in S\) and \(|x - y| < \delta\), then \(g(|x - y|) < \epsilon\), hence \(|f(x) - f(y)| \leq g(|x - y|) < \epsilon\).

Step 5: Conclude uniform continuity of \(f\)

We have shown that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that for every \(x, y \in S\), \(|x - y| < \delta\) implies \(|f(x) - f(y)| < \epsilon\). This confirms that \(f\) is uniformly continuous.

Key Concepts

Continuity

The continuity of a function is a fundamental idea in calculus. In essence, it describes whether small changes in the input of a function lead to small changes in its output.
For a function to be continuous at a point, the function must be defined at that point, the limit of the function as it approaches the point must exist, and the value of the function at the point must equal this limit.
This property makes sure that you can draw a function without lifting your pencil off the paper.

• **Ordinary Continuity** focuses on behavior around individual points.
• **Uniform Continuity**, however, considers the function's behavior globally across its entire domain.
  This means that the same "closeness" criterion works everywhere within the domain.

In the given problem, we see that the function \(g\) is continuous at 0 and \(g(0)=0\). This piece of information is crucial as it acts as a building block to establish the uniform continuity for function \(f\). By understanding this, we can infer that small values for \(x-y\) lead to small values for \(g(|x-y|)\).

Real Analysis

Real analysis is a branch of mathematics that deals primarily with the rigorous study of functions and sequences involving real numbers. It digs deeper into foundational concepts that in calculus are only skimmed over.
In real analysis, we often examine properties like completeness, compactness, and convergence in detail.

• The discipline helps provide a deeper understanding of how functions behave under various conditions, which is essential for proving statements such as uniform continuity.
• In real analysis, understanding the precise definitions and properties of continuity is vital for tackling more complex proofs and theorems.

In our exercise, we're utilizing knowledge from real analysis to establish the uniform continuity of a function. By leveraging the continuity of \(g\) and the inequality given, we use a methodical analytical approach to reach a conclusion.

Function Inequality

Function inequalities are statements indicating that one function is larger or smaller at certain points in comparison to another function.
These inequalities are powerful tools in analysis, aiding in establishing properties like bounds and continuity.

• They provide insight into how two functions relate to each other across different parts of their domains.
• In the context of the problem, the inequality \(|f(x) - f(y)| \leq g(|x-y|)\) is central. This inequality bounds the difference in outputs of function \(f\) using the function \(g\).

The fact that \(g(0)=0\) and \(g\) is continuous at 0 means that for any arbitrarily small positive number, \(g\)'s value can be kept below this amount by choosing \(x\) and \(y\) sufficiently close together. This key insight supports proving that \(f\) is uniformly continuous. By understanding these relationships, we can effectively apply the definitions and arrive at the desired conclusion.