--- title: Continuity description: Continuity is a fundamental property of functions where small changes in the input result in small changes in the output. Learners will understand rigorous definitions using limits, explore uniform continuity, and prove key theorems like the Intermediate Value Theorem. category: mathematics subcategory: real-analysis difficulty: beginner, intermediate, advanced url: /subject/continuity --- # Continuity Continuity is a fundamental property of functions where small changes in the input result in small changes in the output. Learners will understand rigorous definitions using limits, explore uniform continuity, and prove key theorems like the Intermediate Value Theorem. ## Available Resources 1 Videos • 2 Books • 3 Courses • 1 Websites ## Courses ### 1. Real Analysis 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. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLBh2i93oe2quABbNq4I_-hyjhW8eOdgrO **Tags:** real-analysis, continuity, sequences-and-limits, epsilon-delta, proofs ### 2. Real Analysis (MIT 18.100A) **Author:** Casey Rodriguez 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. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ **Tags:** real-analysis, proofs, sequences-and-series, uniform-convergence, riemann-integral ### 3. Introduction to Complex Analysis **Author:** Petra Bonfert-Taylor 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.coursera.org/learn/complex-analysis **Tags:** complex-analysis, analytic-functions, conformal-mapping, contour-integration, residue-theorem ## Videos ### 1. MIT 18.100A Lecture 14: Limits of Functions in Terms of Sequences and Continuity **Author:** Casey Rodriguez 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/resources/18100a-lecture-14-multicam/ **Tags:** real-analysis, continuity, epsilon-delta, sequential-continuity, limits-of-functions ## Books ### 1. Basic Analysis I, Chapter 3: Continuous Functions **Author:** Jiří Lebl 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.jirka.org/ra/html/lim_chapter.html **Tags:** real-analysis, continuity, uniform-continuity, intermediate-value-theorem, open-textbook ### 2. Counterexamples in Analysis **Author:** Bernard R. Gelbaum, John M. H. Olmsted 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. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0486428753?tag=edmonddante07-20 **Tags:** real-analysis, counterexamples, continuity, pathological-functions, proofs ## Websites ### 1. Wolfram MathWorld **Author:** Eric W. Weisstein 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. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/continuity