Real analysis

Real analysis studies the real number system and the limits, continuity, differentiation, integration, and infinite series of real-valued functions.

Real number: A number representing a point on the continuous number line, including rational and irrational numbers. Real analysis takes these numbers as its domain and studies their completeness.
Epsilon–delta definition of a limit: A formal condition stating that function values stay near a target whenever inputs are sufficiently near a point. It gives the precise test for function limits at the heart of analysis.
Intermediate value theorem: A theorem stating that a continuous function on an interval takes every value between its endpoint values. It proves existence of solutions to equations without requiring an explicit formula.
Pointwise convergence: Convergence of a sequence of functions at each point of the domain, with the rate allowed to vary by point. It is weaker than uniform convergence and need not preserve continuity.
Bernard Bolzano: A Czech mathematician and philosopher whose work anticipated rigorous treatments of limits and continuity. His early proofs helped establish standards of rigor later central to analysis.
Completeness of the real numbers: The property that every nonempty set of real numbers bounded above has a least upper bound. Completeness guarantees limits that rational numbers alone cannot supply.
Continuity: A property of a function whose small input changes produce small output changes at each point. Continuity is characterized by limits and preserved under many analytic operations.
Extreme value theorem: A theorem stating that a continuous function on a compact interval attains a maximum and minimum. It justifies optimization claims when continuity and compactness hold.
Uniform convergence: Convergence of functions with one error bound that works at every point of the domain. Unlike pointwise convergence, it preserves continuity under standard conditions.
Augustin-Louis Cauchy: A French mathematician who developed systematic methods for limits, continuity, and convergence. His textbooks helped make limit-based analysis a disciplined subject.

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