Intro Real Analysis, Lec 15: Uniform Continuity, Monotone Functions, Devil's Staircase, Derivatives

Lecture 15.

(0:00) Reminder to review the definition of uniform continuity of a function over an interval and the Uniform Continuity Theorem, as well as other "named" theorems from Chapter 3 of Gordon's text.
(2:35) Brief discussion of linearity as a property of limits and derivatives.
(4:13) Facts to know about monotone functions.
(5:17) The Cantor set and the Devil's Staircase (Cantor's function).
(12:30) Brief review of what to know about variation of a function and what it means for a function to have bounded variation.
(15:28) Derivative of 1/x^2 done with limits in two ways.
(23:34) Derivative of sqrt(x) done with limits.
(28:38) Differentiate a complicated function with various rules (Quotient, Product, and Chain Rules).
(35:35) Abstract form of the Chain Rule.
(37:56) Use the Chain Rule and Product Rule to derive the Quotient Rule.
(40:10) Derive the derivative of the arctangent function arctan(x).
(44:45) Derivation of the derivative of an inverse function.
(46:28) Example calculation of the derivative of an inverse function at a point even when no formula for the inverse function is found.
(50:43) Differentiability implies continuity, so a point of discontinuity will also be a point of non-differentiability.
(51:16) Reminder about there being everywhere continuous nowhere differentiable functions.
(52:08) Vertical tangent line locations are points of non-differentiability as well, such as for cuberoot(x) = x^(1/3) at x = 0.
(53:51) You should be able to prove the Product Rule on exam 2.

Bill Kinney, Bethel University Department of Mathematics and Computer Science. St. Paul, MN.

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[이해준-fs]

Is Cantor Set integrable, maybe lebesque? If so, wouldn't it be 0?

[marcinr]

@Bill Kinney, do you have similar lectures for complex analysis?