What Happened 10/19

On Thursday 10/19, we began class with a weekly situation, “Are we doing mathematics or philosophy?”. Dr. LeCrone then read a thorough and quite verbose definition of a function from another analysis book. We then switched to the topic of the day: continuity, part of Ch. 4 (Continuity on Compact Sets and Uniform Continuity).

We learned the definition of uniform continuity and point-wise continuity. We then looked at three example functions, f(x)= 2x, g(x)=x^2, h(x)=\sqrt(x) all with A=[0,\infty), and examined their graphs, specifically examining V_{\delta_1} (c_1), V_{\delta_2} (c_2), V_{\delta_3} (c_3) and V_{\epsilon_1} (f(c_1)), V_{\epsilon_2} (f(c_2)), V_{\epsilon_3} (f(c_3)). Through this, we were able to visually apply our definition of uniform continuity and check whether each met the definition.

Dr. LeCrone then gave a proof template, and we completed the proof for proving f(x) was continuous on the domain.

Proof template: let \epsilon>0 … (details regarding \delta) … choose \delta= ____ , depending only on \epsilon. Let c \in A be arbitrary and consider x \in A with |x-c| < \delta … (details).. Thus, |f(x)-f(c)|< \epsilon. Since c was arbitrary, we have unif. continuity of f on domain A.

We then completed this proof, using \delta= \epsilon/2

We then constructed a Sequential Criteria for Uniform Continuity and and a Sequential Characterization.

We used this sequential characterization, choosing (x_n)=(n), (y_n)=(n-\frac{1}{n}), and \epsilon_0=1 to show that g(x) is not uniformly continuous.

We observed that h(x) requires us to split the domain in order to characterize, which we will cover on Tuesday.

Finally we covered two theorems. Theorem 4.4.1 states that If $latex f: A \rightarrow R$ continuous and K \subseteq A is compact, then f(k) is compact. We went over a pf of this. Theorem 4.4.2 states that if $latex f: K \rightarrow R$ is continuous on K \subseteq R compact, then f attains a maximum and minimum on K. We then converted this to two equivalent statements, one relating values of f(x) directly and another using supremum and infimum.

Posted on by Elise Favia
Categories: Daily Blogs, What Happened

One Response

  1. Jeremy LeCrone says:
    October 22, 2017 at 10:25 pm

    Thank you for the synopsis from class, Elise. One minor correction in your third paragraph: We showed that f is uniformly continuous on A.

    Oh, and the textbook I read from at the beginning of class is “The Way of Analysis” by Robert Strichartz. It really is a good book, if anyone wants to have a copy in their collection.

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