Unit IV: Real Analysis - Limits and Continuity

Concept of Cluster Point (Limit Point) of a Set in ℝ

The concept of a cluster point (also referred to as a limit point or accumulation point) is a foundational idea in real analysis that allows us to rigorously define the limit of a function at a point, whether or not the function is defined at that specific point.

Definition: Let A be a subset of ℝ. A point c ∈ ℝ is a cluster point of A if every δ-neighborhood V_δ(c) = (c - δ, c + δ) contains at least one point of A different from c.

Symbolically, c is a cluster point of A if for every δ > 0, there exists an x ∈ A such that x ≠ c and |x - c| < δ. This condition can also be expressed using set notation:

(V_δ(c) ∩ A) ∖ {c} ≠ ∅ for all δ > 0

Key Observations and Properties

• Membership: A cluster point c of a set A does not necessarily have to belong to A. For instance, the boundary point of an open interval is a cluster point but is not inside the interval.
• Infinity of Points: If c is a cluster point of A, then every neighborhood of c contains infinitely many points of A.
• Isolated Points: A point c ∈ A that is not a cluster point of A is called an isolated point of A. For an isolated point, there exists a neighborhood that contains no other points of A except c.

Examples of Cluster Points

Common Mistake: Assuming that a finite set can have a cluster point. Finite sets have no cluster points in ℝ because the minimum distance between distinct points is always positive, allowing for the construction of isolated neighborhoods.

Limit of a Function

The limit of a function describes how the function behaves near a target point, rather than at the target point itself.

Definition (ε-δ Criterion): Let A ⊆ ℝ, let f: A → ℝ, and let c be a cluster point of A. We say that a real number L is the limit of f at c if, for every ε > 0, there exists a δ > 0 such that if x ∈ A and 0 < |x - c| < δ, then |f(x) - L| < ε.

This is symbolically written as:

lim_x→c f(x) = L

Step-by-Step Proof Example

Problem: Use the ε-δ definition of a limit to prove that lim_x→2 (3x + 1) = 7.

1. Analyze the target expression: We want to show that for any given ε > 0, there is a δ > 0 such that if 0 < |x - 2| < δ, then |(3x + 1) - 7| < ε.
2. Simplify the inequality:
   |(3x + 1) - 7| = |3x - 6| = 3|x - 2|.
3. Relate δ to ε: We want 3|x - 2| < ε, which is equivalent to |x - 2| < ε/3.
4. Choose δ: Let δ = ε/3.
5. Verify the choice: If 0 < |x - 2| < δ, then:
   |(3x + 1) - 7| = 3|x - 2| < 3δ = 3(ε/3) = ε.
   Thus, the limit is verified.

Sequential Criterion for Limits

The sequential criterion is a powerful equivalence theorem that allows us to use sequence behavior to evaluate and verify function limits.

Theorem (Sequential Criterion for Limits): Let f: A → ℝ and let c be a cluster point of A. The limit lim_x→c f(x) = L if and only if for every sequence (x_n) in A ∖ {c} that converges to c, the sequence of functional values (f(x_n)) converges to L.

This theorem is valuable because it converts questions about function limits into simpler questions about sequence limits, where tools like the Squeeze Theorem for sequences are already established.

Divergence Criteria

To prove that a function does not have a limit at a point, we use divergence criteria. These are formulated by negating the sequential criterion.

Divergence Criterion 1: If f: A → ℝ and c is a cluster point of A, then f does not have a limit L at c if there exists a sequence (x_n) in A ∖ {c} that converges to c, but the sequence (f(x_n)) does not converge to L.

Divergence Criterion 2 (No limit exists at all): f does not have a limit at c if there exist two sequences (x_n) and (y_n) in A ∖ {c} such that both sequences converge to c, but the sequences of functional values (f(x_n)) and (f(y_n)) converge to different limits.

Practical Example: f(x) = sin(1/x) at c = 0

Show that lim_x→0 sin(1/x) does not exist.

1. Select a sequence (x_n) that converges to 0: Let x_n = 1 / (2nπ + π/2). As n → ∞, x_n → 0.
   Calculate the limit of the functional sequence: f(x_n) = sin(2nπ + π/2) = 1 for all n. Thus, lim f(x_n) = 1.
2. Select a second sequence (y_n) that converges to 0: Let y_n = 1 / (2nπ). As n → ∞, y_n → 0.
   Calculate the limit of the functional sequence: f(y_n) = sin(2nπ) = 0 for all n. Thus, lim f(y_n) = 0.
3. Since both sequences converge to 0 but their functional images converge to different limits (1 ≠ 0), the limit does not exist.

Bounded Functions

A function must stay within local bounds near a point if it possesses a limit at that point.

Definition: Let A ⊆ ℝ, f: A → ℝ, and let c be a cluster point of A. The function f is said to be bounded on a neighborhood of c if there exists a δ-neighborhood V_δ(c) of c and a constant M > 0 such that |f(x)| ≤ M for all x ∈ A ∩ V_δ(c).

Theorem on Local Boundedness

Theorem: If a function f: A → ℝ has a limit at c, then f is bounded on some neighborhood of c.

Proof logic: If lim_x→c f(x) = L, choosing ε = 1 ensures there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < 1. By applying the triangle inequality, we have |f(x)| = |f(x) - L + L| ≤ |f(x) - L| + |L| < 1 + |L|. Hence, f is bounded locally by M = max{|f(c)|, |L| + 1}.

Limit Theorems

Algebraic properties of limits allow us to find the limits of complex combinations of functions by evaluating the limits of individual parts.

Theorem (Algebraic Properties): Let A ⊆ ℝ, let c be a cluster point of A, and let f, g: A → ℝ. Suppose that lim_x→c f(x) = L and lim_x→c g(x) = M. Then:

• Sum Rule: lim_x→c (f + g)(x) = L + M
• Difference Rule: lim_x→c (f - g)(x) = L - M
• Product Rule: lim_x→c (fg)(x) = L × M
• Constant Multiple Rule: lim_x→c (kf)(x) = kL for any constant k ∈ ℝ
• Quotient Rule: lim_x→c (f/g)(x) = L/M, provided M ≠ 0

Squeeze Theorem

When direct evaluation of a limit is difficult, we can find it by trapping the function between two helper functions whose limits are known.

Squeeze Theorem: Let A ⊆ ℝ, let c be a cluster point of A, and let f, g, h: A → ℝ. If f(x) ≤ g(x) ≤ h(x) for all x ∈ A, x ≠ c, and if lim_x→c f(x) = L = lim_x→c h(x), then lim_x→c g(x) = L.

Practical Example

Evaluate lim_x→0 x² sin(1/x).

We know that the sine function is bounded between -1 and 1 for all real inputs:

-1 ≤ sin(1/x) ≤ 1

Since x² ≥ 0 for all real x, we can multiply the inequality by x² without changing the direction of the inequality signs:

-x² ≤ x² sin(1/x) ≤ x²

Now evaluate the limits of the bounding functions as x → 0:

lim_x→0 (-x²) = 0 and lim_x→0 (x²) = 0

Because the lower bound and upper bound both approach 0, the Squeeze Theorem guarantees that:

lim_x→0 x² sin(1/x) = 0

Continuous Functions

A function is continuous at a point if its functional value matches its local limit. Visually, this means there are no jumps, holes, or vertical asymptotes at that point.

Definition: Let A ⊆ ℝ, f: A → ℝ, and c ∈ A. The function f is continuous at c if, for every ε > 0, there exists a δ > 0 such that if x ∈ A and |x - c| < δ, then |f(x) - f(c)| < ε.

If c is a cluster point of A, this definition is equivalent to saying:

lim_x→c f(x) = f(c)

If c ∈ A is an isolated point (not a cluster point), then f is automatically continuous at c because there is a δ-neighborhood that contains only the point c itself, making the condition vacuously true.

Comparison: Limits vs. Continuity

Sequential Criterion for Continuity

We can establish the continuity of a function by examining sequences of domain points that converge to the target point.

Theorem: A function f: A → ℝ is continuous at a point c ∈ A if and only if for every sequence (x_n) in A that converges to c, the sequence of functional values (f(x_n)) converges to f(c).

This sequential formulation is highly useful when constructing proofs or showing properties of functions on compact sets.

Discontinuity Criterion

If we need to prove that a function is not continuous at a point, we use the negation of the sequential criterion.

Discontinuity Criterion: Let A ⊆ ℝ, f: A → ℝ, and c ∈ A. The function f is discontinuous at c if and only if there exists a sequence (x_n) in A converging to c such that the sequence of functional values (f(x_n)) does not converge to f(c).

Classic Example: Dirichlet's Function

Consider the function f: ℝ → ℝ defined as:

f(x) = 1 if x is rational, and f(x) = 0 if x is irrational.

We can prove that this function is discontinuous at every real number c:

• Case 1: Let c be rational. Since irrational numbers are dense in ℝ, we can find a sequence of irrationals (x_n) converging to c. The functional values are f(x_n) = 0 for all n, which converges to 0. But f(c) = 1. Since 0 ≠ 1, f is discontinuous at c.
• Case 2: Let c be irrational. Since rational numbers are dense in ℝ, we can find a sequence of rationals (y_n) converging to c. The functional values are f(y_n) = 1 for all n, which converges to 1. But f(c) = 0. Since 1 ≠ 0, f is discontinuous at c.

Combinations of Continuous Functions

Algebraic properties of continuous functions follow directly from the algebraic properties of limits.

Theorem: Let A ⊆ ℝ, f, g: A → ℝ, and c ∈ A. If f and g are continuous at c, then:

• f + g and f - g are continuous at c.
• fg is continuous at c.
• kf is continuous at c for any real constant k.
• f/g is continuous at c, provided that g(c) ≠ 0.

Note: If f and g are continuous on the entire set A, then their sum, difference, product, and scalar multiples are continuous on A. The quotient f/g is continuous on A ∖ {x ∈ A : g(x) = 0}.

Composition of Continuous Functions

The composition of two continuous functions preserves continuity. This means that "a continuous function of a continuous function is continuous."

Theorem: Let A, B ⊆ ℝ, let f: A → ℝ, and let g: B → ℝ such that f(A) ⊆ B. Let c ∈ A. If f is continuous at c and g is continuous at f(c), then the composite function g ∘ f: A → ℝ is continuous at c.

Proof of the Composition Theorem

Let's prove this composition rule using the sequential criterion for continuity:

1. Let (x_n) be any sequence in the domain A such that lim(x_n) = c.
2. Since f is continuous at c, applying the sequential criterion yields:
   lim f(x_n) = f(c).
3. Let y_n = f(x_n) and d = f(c). The sequence (y_n) lies in B and converges to d.
4. Since g is continuous at d = f(c), applying the sequential criterion yields:
   lim g(y_n) = g(d) → lim g(f(x_n)) = g(f(c)).
5. Since this holds for any sequence (x_n) in A converging to c, the composite function g ∘ f is continuous at c.