18.100A Real Analysis (Fall 2020, MIT OCW): Lecture 15 - The Continuity of Sine and Cosine and the Many Discontinuities of Dirichlet's Function

18.100A Real Analysis

18.100A Real Analysis (Fall 2020, MIT OCW). Instructor: Dr. Casey Rodriguez. 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. (from ocw.mit.edu)

Lecture 15 - The Continuity of Sine and Cosine and the Many Discontinuities of ...

Instructor: Dr. Casey Rodriguez. 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.

Go to the Course Home or watch other lectures:

Lecture 01 - Sets, Set Operations and Mathematical Induction

Lecture 02 - Cantor's Theory of Cardinality (Size)

Lecture 03 - Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property

Lecture 04 - The Characterization of the Real Numbers

Lecture 05 - The Archimedean Property, Density of the Rotations, and Absolute Value

Lecture 06 - The Uncountability of the Real Numbers

Lecture 07 - Convergent Sequences of Real Numbers

Lecture 08 - The Squeeze Theorem and Operations Involving Convergent Sequences

Lecture 09 - 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