Day 42, Mat420: Real Analysis

• Announcements:
  • You have an assignment due Wednesday.

  • We decided last time that the final will be entirely in class, and that you'll be able to take it at your convenience during finals week (usual operating hours!) -- with the obvious caveat that you do not discuss it with your peers. That means don't even mention it to each other!

  • Maybe we can get Amber to show us what she did with Jennifer's "image map" connecting theorems, definitions, etc. -- on Wednesday?

  • It is time for course evaluations: please visit eval.nku.edu to give your feedback (and also to make sure that you will get access to your grades earlier).

• Today we continue discussing the material of section 5.1: The Derivative. We do so in the contect of attempting to understand the article about "Pathological Functions" in analysis.
  • Last time we showed that the function in our "Pathological Functions" handout (p. 802) is differentiable at 0. We did this directly, using the definition of the derivative. We then showed that is not differentiable at any other point, by considering well-chosen sequences that converged to another value of (but for which the limits of the difference quotients were not equal).

    Then we proved the pinching theorem of Lemma 2, and so arrived more easily at the result that is differentiable at .

  • Today I want to begin by examining the random function of Figure 5, bounded by , which is also differentiable at , then extend it to the function of figure 6, p. 803. We'll also take a look at the function of figure 7, p. 804.

• Then we want to consider the dyadic rational numbers, and the examples of rather dramatic discontinuity in the rest of the paper.

Links:

• "Pathological Functions" handout: a Nov. 2011 Monthly article.