Lecture 17: Uniform Continuity and the Definition of the Derivative

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MIT 18.100A Real Analysis, Fall 2020
Instructor: Dr. Casey Rodriguez

We wrap up our current study of continuous functions by considering uniform continuity. We show that uniform continuity is equivalent to continuity on a closed and bounded interval, and begin to consider the derivative of a function.

License: Creative Commons BY-NC-SA

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Can't believe he mentioned Borat, I love those little tidbits during these lectures.

nathanielthomas

Love the proof of the power rule! It was always so mysterious back in calculus, I thought it was actually empirical when I first learned about it. Great to see it in concrete terms!

nathanielthomas

Finally, the almighty derivative, I've been waiting for this!

nathanielthomas

Hey self-learners! Prof. Rodriguez made a conclusion at 27:40 that may cause confusion. He said "something <= epsilon" when he was trying to prove "something < epsilon". That is totally correct, given that proposition "something = epsilon" is cleary false. Given both proposition, he can conclude that "something < epsilon" is true.

That form of logical procedure is called disjunctive syllogism. One of its forms is the following.
Let A and B propositions. The rule says ( (A or B) and ¬B ) ---> A is correct.

Example:
Let A = "10 is less than 100", B = "10 is equal to 100". By disjunctive syllogism, if I have statements "10 is less or equal to 100" AND "10 is NOT equal to 100" then I can conclude "10 is less than 100".

nicolasg.b.

Very helpful lesson. Thank you Professor..🙏🙏

VaibhavSharma-zjgk

at 27:40 I believe there's a typo: you should have a "<" rather than leq for |1+1| *delta

hakus

Why at around 1:09 we can change the start and the end of the summation?

Daniel-kkkr

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