Problem 16 - Basic Analysis: Introduction to Real Analysis

Question

Let \(\left(X, d_{X}\right)\) and \(\left(Y, d_{y}\right)\) be metric spaces, \(S \subset X, p \in X\) a cluster point of \(S,\) and let \(f: S \rightarrow Y\) be a function. Prove that \(f: S \rightarrow Y\) is continuous at \(p\) if and only if $$ \lim _{x \rightarrow p} f(x)=f(p) $$

Step-by-Step Solution

Answer

The function \( f \) is continuous at \( p \) if and only if \( \lim_{x \to p} f(x) = f(p) \).

Step 1: Define Continuity at a Point

To prove the statement, we start by defining what it means for a function \( f: S \rightarrow Y \) to be continuous at a point \( p \). A function \( f \) is continuous at a point \( p \) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x \in S \), if \( 0 < d_X(x, p) < \delta \), then \( d_Y(f(x), f(p)) < \epsilon \).

Step 2: Define the Limit Condition

The statement \( \lim_{x \to p} f(x) = f(p) \) means that for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for all \( x \in S \setminus \{p\} \), if \( d_X(x, p) < \delta \), then \( d_Y(f(x), f(p)) < \epsilon \). This aligns with the definition of continuity at \( p \).

Step 3: Prove the 'If' Direction

Assume that \( \lim_{x \to p} f(x) = f(p) \). By definition, for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( x \in S \setminus \{p\} \) and \( d_X(x, p) < \delta \), it follows that \( d_Y(f(x), f(p)) < \epsilon \). This matches the condition for continuity at \( p \), establishing that \( f \) is continuous at \( p \).

Step 4: Prove the 'Only If' Direction

Assume that \( f \) is continuous at \( p \). By the definition of continuity, for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( x \in S \) and \( 0 < d_X(x, p) < \delta \), then \( d_Y(f(x), f(p)) < \epsilon \). Since continuity already implies the limit condition for points \( x \in S \setminus \{p\} \), it follows that \( \lim_{x \to p} f(x) = f(p) \).

Step 5: Conclusion

Both directions have been proved, showing that \( f \) is continuous at \( p \) if and only if \( \lim_{x \to p} f(x) = f(p) \). Thus, the equivalence of continuity at a point and the limit condition is established.

Key Concepts

Understanding Metric Spaces

To fully grasp the concept of metric spaces, it's essential to understand what they are. A metric space is a set equipped with a metric, which is a function that defines a distance between any two elements of the set. Think of it like a framework to measure how far apart two points are from each other. In mathematical terms, a metric space consists of a set \(X\) and a metric \(d_X(x, y)\) that satisfies the following conditions for all points \(x, y, z \in X\):

• \(d_X(x, y) \geq 0\) (non-negativity) with \(d_X(x, y) = 0\) if and only if \(x = y\).
• \(d_X(x, y) = d_X(y, x)\) (symmetry).
• \(d_X(x, z) \leq d_X(x, y) + d_X(y, z)\) (triangle inequality).

In simpler terms, a metric space allows us to "measure" the proximity or distance consistently and logically across various elements within a set.

Limit of a Function

The limit of a function at a given point captures the idea of a function approaching a specific value as the input approaches a certain point. Formally, suppose we have a function \(f: S \rightarrow Y\) and a point \(p \in X\). We say the limit of \(f(x)\) as \(x\) approaches \(p\) is \(f(p)\), denoted as \(\lim_{x \to p} f(x) = f(p)\), if the function value \(f(x)\) gets arbitrarily close to \(f(p)\) as \(x\) gets arbitrarily close to \(p\).
This is formally defined such that for every \(\epsilon > 0\), there exists a \(\delta > 0\) for which:

• If \(0 < d_X(x, p) < \delta\), then \(d_Y(f(x), f(p)) < \epsilon\).

The idea of limits is crucial in calculus and mathematical analysis because it lays the groundwork for understanding continuity and convergence behavior in functions.

Understanding Cluster Points

A cluster point is a concept used to describe a type of "limit point" for a set. Given a subset \(S \subset X\) in a metric space, a point \(p \in X\) is considered a cluster point of \(S\) if every neighborhood around \(p\) contains at least one point from \(S\) different from \(p\) itself. This implies that \(p\) can be "approached" by points from \(S\).
To visualize this, think of the concept in terms of proximity:

• No matter how small an epsilon-neighborhood you take around \(p\), there will always be some point from \(S\) within this neighborhood.
• This defines \(p\)'s characteristic as a cluster point because points in \(S\) seemingly "gather" around \(p\).

Cluster points are vital when discussing continuity and limits because they help us identify where function behavior reflects changes in its input near specific regions.