MIT 18.100A Real Analysis, Fall 2020

Instructor: Dr. Casey Rodriguez View the complete course: http://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts through a study of real numbers, and teaches an understanding and construction of proofs. License: Creative Commons BY-NC-SA More information at https://ocw.mit.edu/terms More courses at https://ocw.mit.edu Support OCW at http://ow.ly/a1If50zVRlQ We encourage constructive comments and discussion on OCW’s YouTube and other social media channels. Personal attacks, hate speech, trolling, and inappropriate comments are not allowed and may be removed. More details at https://ocw.mit.edu/comments.

Curated by: MIT OpenCourseWare (25 videos)

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Currently Playing: Lecture 15: The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet's Function

MIT 18.100A Real Analysis, Fall 2020 Instructor: Dr. Casey Rodriguez View the complete course: http://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP61O7HkcF7UImpM0cR_L2gSw We wrap up our discussion of cluster points and move towards continuity. We show that sine and cosine are continuous, prove facts about algebraic operations on continuous functions, and define the Dirichlet function. License: Creative Commons BY-NC-SA More information at https://ocw.mit.edu/terms More courses at https://ocw.mit.edu Support OCW at http://ow.ly/a1If50zVRlQ We encourage constructive comments and discussion on OCW’s YouTube and other social media channels. Personal attacks, hate speech, trolling, and inappropriate comments are not allowed and may be removed. More details at https://ocw.mit.edu/comments.

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Tracks in this Playlist

• Lecture 1: Sets, Set Operations and Mathematical Induction
• Lecture 2: Cantor's Theory of Cardinality (Size)
• Lecture 3: Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property
• Lecture 4: The Characterization of the Real Numbers
• Lecture 5: The Archimedian Property, Density of the Rationals, and Absolute Value
• Lecture 6: The Uncountabality of the Real Numbers
• Lecture 7: Convergent Sequences of Real Numbers
• Lecture 8: The Squeeze Theorem and Operations Involving Convergent Sequences
• Lecture 9: Limsup, Liminf, and the Bolzano-Weierstrass Theorem
• Lecture 10: The Completeness of the Real Numbers and Basic Properties of Infinite Series
• Lecture 11: Absolute Convergence and the Comparison Test for Series
• Lecture 12: The Ratio, Root, and Alternating Series Tests
• Lecture 13: Limits of Functions
• Lecture 14: Limits of Functions in Terms of Sequences and Continuity
• Lecture 15: The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet's Function
• Lecture 16: The Min/Max Theorem and Bolzano's Intermediate Value Theorem
• Lecture 17: Uniform Continuity and the Definition of the Derivative
• Lecture 18: Weierstrass's Example of a Continuous and Nowhere Differentiable Function
• Lecture 19: Differentiation Rules, Rolle's Theorem, and the Mean Value Theorem
• Lecture 20: Taylor's Theorem and the Definition of Riemann Sums
• Lecture 21: The Riemann Integral of a Continuous Function
• Lecture 22: Fundamental Theorem of Calculus, Integration by Parts, and Change of Variable Formula
• Lecture 23: Pointwise and Uniform Convergence of Sequences of Functions
• Lecture 24: Uniform Convergence, the Weierstrass M-Test, and Interchanging Limits
• Lecture 25: Power Series and the Weierstrass Approximation Theorem