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Real Analysis

Real Analysis is the branch of mathematical analysis dealing with the real numbers and real-valued function of a real variable. It rigorously establishes the properties of limits, continuity, differentiation, and integration.

Key Concepts

• Continuity and Limits: Study of limit (mathematics) and continuity as foundational tools.
• Differentiation: Rigorous definition of the derivative and its properties.
• Integration: Construction of the Riemann integral and Lebesgue integral.
• Sequences and Series: Convergence criteria for sequence and series of real numbers.
• Topology of $\mathbb{R}$: Properties of open/closed sets, compactness, and connectedness.

Pathological Functions

Real analysis often explores functions that defy intuitive geometric visualization, serving as counterexamples to common conjectures.

• Conway’s Base 13 Function: A function $f:\mathbb{R}\to \mathbb{R}$ that maps every non-empty open interval onto the entire real line $\mathbb{R}$.
  • Despite being discontinuity everywhere, it satisfies the intermediate value theorem on every interval.
  • It is constructed using base-13 expansions of real numbers to encode arbitrary real values.
  • See Conway’s Base 13 Function: Explaining its Unique Mathematical Definition for a detailed breakdown of its construction and implications.
  • Video resource: Conway’s Base 13 Function: Explaining its Unique Mathematical Definition
• Weierstrass Function: A classic example of a function that is continuous function everywhere but differentiable nowhere.
• Dirichlet Function: The indicator function of the rational number, which is Lebesgue integrable but not Riemann integrable.

Related Topics

• Complex Analysis
• Measure Theory
• Functional Analysis
• Topology