COALITION MATH 152, SPRING 1998

              (S. A. Fulling, assisted by Vera Rice)

                             Day 26.R


    1.  Discussion of "Everybody loves a lover" (postponed from 
    last time).

    2.  Definitions of limit and of continuity (the latter 
    reviewed from last semester).  It took the greatest 
    mathematicians 200 years to understand this [the epsilon 
    definition], so you shouldn't get upset if you have a little 
    trouble with it at first.

    3.  The epsilon-delta games.  (The statements in question are 
    written out in logical/algebraic notation on transparencies 
    beforehand.) 

        A.  Let's see that 1/x is continuous at x = 1.

            i.  Team A chooses an epsilon.

           ii.  Team B chooses a delta (trying to make the 
            statement TRUE).

          iii.  Team A chooses an x (trying to make the statement 
            FALSE).

           iv.  Reach consensus that Team B has won the round (or 
            help them to fix their delta if defective).

            v.  Team A may choose a smaller epsilon ... or give 
            up.

        B.  Change the rules:  Put the delta quantifier first.

            i.  Team B chooses a delta.

           ii.  Team A chooses epsilon and x (trying to make the 
            statement FALSE).

          iii.  Team B may choose a smaller delta ... or give up.

           iv.  Note that this type of statement is true for 
            CONSTANT functions only (Ex. 10 of logic handout).
 
        C.  New game with new teams:  Show that 1/x is continuous 
        at ALL x > 0.  ("For every a and every epsilon there is a 
        delta such that for every x ...")

            i.  Team A chooses a and epsilon.

           ii.  Team B chooses delta (trying to make the 
            statement TRUE).
   
          iii.  Team A chooses x (trying to make the statement 
            FALSE). 

           iv.  Team A may choose a different x and epsilon ... 
            or give up.

        D.  Change the rules:  Put the "a" quantifier inside the 
        delta one.

            i.  Team A chooses epsilon.

           ii.  Team B chooses delta. 
    
          iii.  Team A chooses a and x.

           iv.  Team B may choose a smaller delta ... or give up.

            v.  Can we find a function that lets Team B win this 
            game?  Such a function is UNIFORMLY CONTINUOUS (Ex. 
            10 of logic handout).
NOT SURE HOW MUCH THE STUDENTS GOT OUT OF THIS.   MORE 
PRELIMINARY (OR SIMULTANEOUS) THEORETICAL EXPLANATION MIGHT HAVE 
BEEN HELPFUL, BUT WE WERE BEHIND SCHEDULE.

    4.  Show examples (from old tests) of acceptable student 
    proofs of limit statements for linear functions.