Real Analysis
The Bright Side of Mathematics
Free video course from The Bright Side of Mathematics, in short numbered episodes, on sequences, limits, series, continuity, differentiability and integration. Learners finish able to read and write epsilon-delta arguments and prove standard results about continuous functions.
More resources on Continuity
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Real Analysis (MIT 18.100A)
Rigorous foundations of calculus: real numbers, convergence of sequences and series, continuity, differentiability, the Riemann integral, sequences and series of functions, uniform convergence and interchange of limits. Includes 25 lecture videos, notes, problem sets and exams, and teaches how to read and construct proofs.
MIT 18.100A Lecture 14: Limits of Functions in Terms of Sequences and Continuity
MIT lecture defining continuity via epsilon-delta and sequences. It opens a five-lecture run (Lectures 13-17) on function limits, continuity of sine and cosine, Dirichlet's function, the min/max and intermediate value theorems, and uniform continuity.
Basic Analysis I, Chapter 3: Continuous Functions
Free open textbook chapter covering limits of functions, epsilon-delta and sequential continuity, the extreme and intermediate value theorems, uniform continuity, limits at infinity, and monotone functions, with full proofs and graded exercises.
Counterexamples in Analysis
Collection of counterexamples in real analysis: pathological functions, sequences, series, sets and integrals that show where standard theorems on continuity, differentiability, integration and convergence fail. Helps the reader test conjectures and understand why each hypothesis in a theorem is needed.
Introduction to Complex Analysis
Wesleyan University course on complex analysis covering complex numbers, analytic functions, conformal mappings, Möbius transformations, contour integration, Cauchy's theorem, power and Laurent series, and residues. Learners finish able to evaluate contour integrals and analyze complex-differentiable functions.