Week 5.

Solutions to homework 3.

Homework due Tuesday, July 20, 2004 (homework 4): 3.3) 1, 2, 7, 9, 17, 19.

Monday, July 19, 2004. We started to discuss continuity and some strange examples of how functions can fail to be continuous. We saw an example of a function that was continuous at 0 but discontinuous at every other real number; and an example of a function which was continuous at every irrational number but discontinuous at every rational number. We started to discuss the proof of the Intermediate Value Property for continuous functions.

Tuesday, July 20, 2004. We proved that if f is continuous on [a,b], then f has the Intermediate Value Property on [a,b]. We started to discuss a partial converse to this: if f is piecewise monotonic on [a,b] and has the Intermediate Value Property on [a,b], then f is continuous on (a,b) (and in fact it's continuous on [a,b] given the obvious definition of continuity at an end point).

Wednesday, July 21, 2004. We proved a partial converse of the Intermediate Value Theorem: if f has the Intermediate Value Property on [a,b] and is piecewise monotonic on [a,b], then f is continuous on [a,b]. We proved that sums and products of continuous functions are continuous. Remember we'll have an extra class on Thursday, 2:30-3:18.

Thursday, July 22, 2004 (together with extra class 2:30-3:18). We proved that compositions of continuous functions are continuous, and if c is not zero and f is continuous at c, then 1/f is continuous at c also. We noted that it is not necessarily true that the sum of two functions with the intermediate value property will have the intermediate value property (there is an example in the book on page 92). We proved that if f is differentiable at c, then f is continuous at c. Starting section 3.5, we proved that a continuous function on a closed bounded interval is bounded. We would like to prove that it actually achieves its bounds somewhere on the interval; to do that we introduced the concept of least upper bound and greatest lower bound. We proved that in the real numbers, any (non-empty) set which has an upper bound must have a least upper bound. (And we noted this property fails to hold if we restrict our universe to rational numbers). To prove this we used the nested interval principle; in fact the least upper bound property is equivalent to the nested interval principle.

Friday, July 23, 2004. We proved that a continuous function on a closed bounded interval actually achieves its maximum and minimum values at some points in [a,b]. We proved the Fermat Theorem on extrema (if f has an extremum at c in (a,b) and f'(c) exists, then f'(c)=0). We discussed an example of a function that was differentiable only at the origin, and an example of a function where f'(0) is positive but f is not monotonically increasing in any interval containing 0. Finally we proved Rolle's Theorem.