2301490 SEMINAR Abstract Outline

Lemma3
Lemma4
Lemma5
Theorem2
: If f is bounded on : {a ≤ x ≤ b} , and c is an intermediate point,
then ∫ f(x)dx
and
∫ f(x)dx
= ∫ f(x)dx
= ∫ f(x)dx
+ ∫ f(x)dx
+ ∫ f(x)dx
: Suppose that f is bounded on I: {a ≤ x ≤ b}
Let F and G be defined on I by
F(x) ≡ ∫ f(t)dt and G(x) =
∫ f(t)dt , {a < x ≤ b} , and F(a)=G(a)=0.
Then F'(x )=G'(x )=f (x ) at each point x of I at which f is continuous.
(At a and b, the derivatives are of course one-sided )
: If a, b ∈ R such that |a − b| ≤ ε for all ε > 0, then a=b
: If f: [a, b] → R is bounded, and continuous except on a set of measure zero, then f is Riemann
integrable on [a, b]