Understanding Analysis

1. The Real Numbers: More Than Just Rationals

“[It] is a ‘simple’ theorem, simple both in idea and execution, but there is no doubt at all about [it being] of the highest class. [It] is as fresh and significant as when it was discovered—two thousand years have not written a wrinkle on [it].”

Beyond the Rational. The discovery that √2 is irrational shattered the ancient Greek understanding of numbers, revealing that the rational numbers (fractions) are insufficient to describe all lengths. This necessitates an expansion to the real numbers, which include both rational and irrational numbers, filling in the "gaps" on the number line.

• The set of natural numbers (N) is extended to integers (Z), then to rational numbers (Q), and finally to real numbers (R).
• The real numbers form a field, meaning they support addition, subtraction, multiplication, and division.
• The real numbers are ordered, allowing us to compare their magnitudes.

Completeness is Key. The real numbers are distinguished from the rationals by the Axiom of Completeness, which states that every nonempty set of real numbers that is bounded above has a least upper bound. This property ensures that there are no "holes" in the real number line, allowing us to take limits and perform other operations that are not possible with the rationals alone. The real numbers are the foundation for calculus and analysis.

Irrational Numbers Abound. While we often encounter rational numbers, the set of irrational numbers is actually much larger. The irrationals are also densely packed on the number line, meaning that between any two real numbers, there is always an irrational number. This highlights the complexity and richness of the real number system.

2. Limits: The Foundation of Analysis

A sequence (an) converges to a real number a if, for every positive number ε, there exists an N ∈ N such that whenever n ≥ N it follows that |an − a| < ε.

Precise Definition. The concept of a limit is made rigorous using the epsilon-delta definition, which provides a precise way to describe the behavior of a sequence as it approaches a particular value. This definition is the cornerstone of analysis, allowing us to move beyond intuitive notions and establish mathematical truths.

• The definition involves a challenge (ε) and a response (N).
• The value of N depends on the choice of ε.
• The smaller the ε, the larger N may have to be.

Quantifiers are Crucial. The definition of a limit relies heavily on the use of quantifiers like "for all" and "there exists." The order of these quantifiers is critical, and understanding how to manipulate them is essential for writing rigorous proofs.

• "For all ε > 0" means the statement must hold for every possible choice of ε.
• "There exists N ∈ N" means that we must be able to find at least one suitable N.

Divergence. A sequence that does not converge is said to diverge. To prove that a sequence does not converge to a particular value, we must demonstrate that there is at least one ε for which no suitable N can be found. This highlights the importance of understanding the logical negation of the definition of convergence.

3. Continuity: A Delicate Balance

A function f : A → R is continuous at a point c ∈ A if, for all ε > 0, there exists a δ > 0 such that whenever |x − c| < δ (and x ∈ A) it follows that |f(x) − f(c)| < ε.

Local Behavior. Continuity is a local property, meaning that it is defined at a particular point in the domain of a function. A function is continuous at a point c if the values of the function get arbitrarily close to f(c) as x gets arbitrarily close to c.

• The definition of continuity is similar to the definition of a limit, but it requires that the point c be in the domain of the function.
• The value of δ depends on the choice of ε and the point c.

Sequential Characterization. A function is continuous at a point c if and only if for every sequence (xn) that converges to c, the sequence (f(xn)) converges to f(c). This sequential characterization of continuity is often useful for proving that a function is not continuous.

• If a sequence (xn) converges to c, but (f(xn)) does not converge to f(c), then f is not continuous at c.

Discontinuities. Functions can have different types of discontinuities, including removable discontinuities, jump discontinuities, and essential discontinuities. The nature of these discontinuities is important for understanding the behavior of functions and their derivatives.

4. Differentiation: Slopes and More

The limit of a sequence, when it exists, must be unique.

The Derivative as a Limit. The derivative of a function at a point is defined as the limit of a difference quotient. This limit represents the slope of the tangent line to the graph of the function at that point.

• The derivative is a local property, defined at a particular point in the domain.
• The existence of the derivative implies the continuity of the function at that point.

Algebraic Properties. Differentiable functions behave well with respect to algebraic operations. The derivative of a sum, product, or quotient of differentiable functions can be computed using familiar rules.

• The Chain Rule provides a formula for the derivative of a composition of differentiable functions.

Derivatives Need Not Be Continuous. Although differentiability implies continuity, the derivative function itself is not always continuous. The function x²sin(1/x) is a classic example of a differentiable function whose derivative is not continuous at the origin. This highlights the subtle relationship between continuity and differentiability.

5. Integration: Area and Beyond

A sequence that does not converge is said to diverge.

Riemann Sums. The Riemann integral is defined using upper and lower sums, which are approximations of the area under a curve using rectangles. The integral exists if the upper and lower sums converge to the same value as the width of the rectangles tends to zero.

• The upper sum is an overestimate of the area, while the lower sum is an underestimate.
• The Riemann integral is defined independently of differentiation.

Integrability Criterion. A bounded function is Riemann-integrable if and only if, for every ε > 0, there exists a partition of the domain such that the difference between the upper and lower sums is less than ε. This criterion provides a rigorous way to determine whether a function is integrable.

• Continuous functions are always Riemann-integrable.
• Functions with a finite number of discontinuities are also Riemann-integrable.

The Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus establishes the inverse relationship between differentiation and integration. It states that the derivative of an integral is the original function, and the integral of a derivative can be computed using the antiderivative. This theorem is a cornerstone of calculus and analysis.

6. Sequences and Series of Functions: Infinite Power

Every convergent sequence is bounded.

Pointwise vs. Uniform Convergence. A sequence of functions converges pointwise if, for each point in the domain, the sequence of function values converges. Uniform convergence is a stronger condition, requiring that the convergence be "equally fast" at all points in the domain.

• Pointwise convergence does not guarantee that the limit function will inherit properties like continuity or differentiability.
• Uniform convergence is often required to ensure that the limit function has the same properties as the functions in the sequence.

Uniform Convergence and Continuity. The uniform limit of continuous functions is continuous. This result is a powerful tool for proving the continuity of functions defined as infinite sums or limits.

• The Continuous Limit Theorem states that if a sequence of continuous functions converges uniformly, then the limit function is also continuous.

Uniform Convergence and Differentiation. If a sequence of differentiable functions converges pointwise, and the sequence of derivatives converges uniformly, then the limit function is differentiable, and its derivative is the limit of the derivatives. This result is crucial for justifying term-by-term differentiation of infinite series.

7. The Interplay of Continuity, Differentiation, and Integration

The limit of a sequence, when it exists, must be unique.

Derivatives Need Not Be Continuous. Although differentiability implies continuity, the derivative function itself is not always continuous. The function x²sin(1/x) is a classic example of a differentiable function whose derivative is not continuous at the origin.

• This highlights the subtle relationship between continuity and differentiability.

Darboux's Theorem. Although derivatives need not be continuous, they do possess the intermediate value property. This means that if a derivative function attains two distinct values, it must also attain every value in between.

• This property is a consequence of the Mean Value Theorem.

Lebesgue's Theorem. A bounded function is Riemann-integrable if and only if its set of discontinuities has measure zero. This result provides a complete characterization of the class of Riemann-integrable functions.

• The set of discontinuities of a Riemann-integrable function must be "small" in a precise mathematical sense.

8. Topology: The Shape of the Real Line

The limit of a sequence, when it exists, must be unique.

Open and Closed Sets. Open sets are defined by the property that every point in the set has a neighborhood that is also contained in the set. Closed sets are defined as sets that contain all of their limit points.

• Open and closed sets are not antonyms.
• The complement of an open set is closed, and vice versa.

Compact Sets. Compact sets are sets where every sequence has a convergent subsequence that converges to a limit in the set. In R, compact sets are precisely the sets that are both closed and bounded.

• Compact sets are important because they allow us to extend results that are true for finite sets to infinite sets.

Connected Sets. Connected sets are sets that cannot be separated into two disjoint open sets. In R, connected sets are precisely the intervals.

• Continuous functions map connected sets to connected sets.

Perfect Sets. A perfect set is a closed set that contains no isolated points. The Cantor set is a classic example of a perfect set.

• Perfect sets are always uncountable.

9. Cardinality: Comparing Infinities

The set A has the same cardinality as B if there exists f : A → B that is 1–1 and onto.

One-to-One Correspondence. The cardinality of a set refers to its size. Two sets have the same cardinality if there exists a one-to-one and onto function between them. This definition allows us to compare the sizes of infinite sets.

• A function is one-to-one if no two elements of the domain map to the same element of the range.
• A function is onto if every element of the range is mapped to by at least one element of the domain.

Countable Sets. A set is countable if it has the same cardinality as the natural numbers. The integers and the rational numbers are countable sets.

• Countable sets can be arranged into an infinitely long list.

Uncountable Sets. The real numbers are uncountable, meaning that they have a larger cardinality than the natural numbers. The power set of any set has a larger cardinality than the set itself.

• The set of irrational numbers is also uncountable.
• The set of all subsets of the natural numbers is uncountable.

10. The Power of Abstraction: Metric Spaces

The limit of a sequence, when it exists, must be unique.

Generalizing Distance. A metric space is a set together with a function that defines a notion of distance between any two elements in the set. This allows us to extend concepts from R to more general settings.

• A metric must satisfy the triangle inequality.
• The Euclidean distance is a metric on R2.
• The sup norm is a metric on the space of continuous functions.

Convergence and Completeness. The definitions of convergence and Cauchy sequences can be extended to metric spaces. A metric space is complete if every Cauchy sequence converges to an element in the space.

• The real numbers are a complete metric space.
• The space of continuous functions on [0,1] is also a complete metric space.

Baire Category Theorem. The Baire Category Theorem states that a complete metric space cannot be written as the countable union of nowhere-dense sets. This result has important implications for the study of function spaces.

• The set of continuous functions that are differentiable at even one point is a meager set in the space of all continuous functions.

Last updated: February 2, 2025