2301490 SEMINAR Abstract Outline

2301490 SEMINAR
Title : An Elementary Proof That a Bounded a.e. Continuous Function is Riemann Integrable
Speaker : Akkaporn Wongpradit
Room : TAB 229
Date : Tuesday 23 August, 2011.
Time : 14.00-15.00
Reference : The American Mathematical Monthly, Vol. 95, No. 3 (Mar., 1988), pp. 249-252
Abstract
From calculus I, we know that every continuous function over a closed interval is Riemann
integrable. This article weakens the condition by considering a bounded function and a.e. continuous function over
an closed interval. With these condition on f, we can prove that f is Riemann integrable
Outline
Definition1 : f: [a, b] → R is continuous a.e. if and only if f is continuous except on a set of measure zero
Definition2 (Riemann Integrability)
: A bounded function f defined on the interval [a, b] is Riemann-integrable
if ∫ f(x)dx = ∫ f(x)dx. In this case, we define ∫ f or
∫ f(x)dx to be this common value;
namely,∫ f
.
=
∫ f(x)dx = ∫ f(x)dx
Theorem1 : If f is continuous on [a, b], then it is Riemann integrable there.
Definition3 : A set E of real numbers has measure zero if for each ε > 0 there is a finite or infinite
sequence {I } of open intervals covering E and satisfying ∑ |I |
of I
Lipschitz condition
: f(x) satisfies a Lipschitz condition on an interval I if there exists M > 0 such that
|f(x) − f(y)| ≤ M|x − y| for all x, y ∈ I with x ≠ y
≤ ε where |I | is the length
Lemma1 : If f: [a, b] → R satisfies a Lipschitz condition and f′(x) = 0 except on a set of measure zero,
then f is a constant function.
Lemma2 : If f is bounded on : {a ≤ x ≤ b} , and if M and m are upper and lower bounds for f on I,
then m(b − a) ≤ ∫ f(x)dx ≤ ∫ f(x)dx ≤ M(b − a)

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]