Lecture , Friday 1999/12/03, 8:00-9:50 AM:

1-d calculus, 
Counter examples in 1-d Calculus:
f:R -> R bounded, f'(x) -> 0 as x -> \infty
Is it true that lim f(x) exist?

f(x) -> 0 as x -> \infty
is it true that  f'(x) -> 0 as x -> \infty?

\int_0^\infty f(x) converge, 
is it true that  f(x) -> 0 as x -> \infty?      

Riemann integral on [a,b]
definition of U, L 
lemma on U(P) >= U(P') if P' is a refinement of P.
definition of uppper and lower integrals.
example of nonintegrable functions.

Thm: f bdd cont except finitely many point => f integrable.
proof: case 1: f cont on [a,b] hence unoforly cont. done.
       case 2: f cont on (a,b) devide into [a,a'], [a',b'], [b',b] 
	       thus [a',b'] redices to case 1 and let a'-> a, b' -> b
       case 3: f discont at a1, a2, ... an. 
                add them into the partition and devise into 
		[a,a1],[a1,a2],[a2,a3],...[an,b]

Lecture , Tuesday 1999/12/07, 8:00-9:50 AM:
Remarked on iff condition for Riemann integrability.

Explained measure zero set. eg: finite set, countable set, cantor set.
explained why Cantor set is uncountable.

How to define inproper integrals.

Uniform convergence:
definition. example, f(x) = x^n on (0,1).

Theorem and proof: unofotm limit of cont function is cont.

Eg: fn -> f point wise, int fn not -> int f
fn = (n+1) x^n
Lecture , Friday 1999/12/10, 8:00-9:50 AM:
unoform contiuous and the supnorm.

Weirstrass M test

interchange of lim, or sum with int of d/dx over [a,b]

space of cont fns.

fn converge uniformly <=> fn is Cauchy in supnorm.

Cb(A,N) is complete.

Lecture , Tuesday 1999/12/14, 8:00-9:50 AM:
Topology on C_b(A,N), metric, open set, compact sets.
iExamples of nonequicont sets.
: fn(x) = nx, x^n, zigzag, x^(1/n),
| f'_n | <= M  then equicont, but not conversely.