Real Analysis
The study of real numbers, sequences, series, limits, continuity, differentiation, and integration on the real number line.
Sequences and Series
A sequence is an ordered list of numbers; a series is the sum of a sequence. Convergence is central: a sequence converges if its terms approach a finite limit. A series converges if its partial sums approach a finite value. Key tests include the ratio test, root test, comparison test, and integral test.
Continuity and Differentiability
A function f is continuous at a point if the limit equals the function value. It is uniformly continuous if the same δ works for all points. Differentiability implies continuity, but not vice versa. The mean value theorem states that for a differentiable function on [a,b], there exists c where f'(c) equals the average rate of change.
Metric Spaces
A metric space is a set equipped with a distance function satisfying non-negativity, symmetry, and the triangle inequality. Metric spaces generalize the notion of distance and provide the framework for analysis in abstract settings. Completeness (every Cauchy sequence converges) is a key property; complete metric spaces are called Banach spaces when they are also normed vector spaces.
Measure Theory
Measure theory provides a rigorous foundation for integration. A measure assigns a non-negative number to subsets of a space, generalizing length, area, and volume. The Lebesgue integral extends the Riemann integral to a broader class of functions and is essential for probability theory and functional analysis.