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Michael11
science forum Guru Wannabe

Joined: 15 Aug 2005
Posts: 103

Posted: Tue Jul 18, 2006 7:54 pm    Post subject: please recommend good reference book on mathematics?

HI all,

I often met with confusion about some theorems, concepts, definitions, etc.
that I really need a reference book on my bookshelf that I can refer to from
time to time.

For example, in reading a text I came cross the following questions
recently:

1. In a text book, "Riemann integral can be defined for continuous functions
and can be extended to functions with finitely many discontinuities".

I am not sure if this "finitely many" means finite, or countable?

Riemann integral should be okay for functions with countably many
discontinuities, right?

2. "A necessary and sufficient condition for a function f to be Riemann
integrable on a finite closed interval [a, b] is that f is bounded on [a, b]
and almost everywhere continuous on [a, b]".

In this theorem, can "f is bounded on [a, b]" be changed into "f is almost
everywhere bounded on [a, b]"?

3. "If f is continuously differentiable on a finite interval..."

Here: "continuously differententiable" means the function itself is
continuous and differentiable or the function's derivative is continuous?

4. "A finite variation function on [a, b] is differentiable almost
everywhere on [a, b]"...

5. This is my own statement: "monotonic functions always have left and right
derivatives, but their left and right derivatives are not neccessarily
equal." Is this statement correct?

How to prove this?

--------------------------------------

The above are just examples of questions I had in mind while reading a
stochastic calculus book. Since this book is not on analysis per se, it does
not give detailed account about the proof and logics behind statements...

As you can see I really need "reference books" that can be put on my
bookshelf that I can referred to from time to time.

Can anybody give me some pointers?

Thanks a lot!!!
Ronald Bruck
science forum Guru

Joined: 05 Jun 2005
Posts: 356

Posted: Tue Jul 18, 2006 8:41 pm    Post subject: Re: please recommend good reference book on mathematics?

In article <e9je8f\$3jp\$1@news.Stanford.EDU>, Michael
<michael.monkey.in.the.jungle@gmail.com> wrote:

 Quote: HI all, I often met with confusion about some theorems, concepts, definitions, etc. that I really need a reference book on my bookshelf that I can refer to from time to time. For example, in reading a text I came cross the following questions recently: 1. In a text book, "Riemann integral can be defined for continuous functions and can be extended to functions with finitely many discontinuities". I am not sure if this "finitely many" means finite, or countable? Riemann integral should be okay for functions with countably many discontinuities, right?

It says finite, not countable. And it MEANS finite. No, a bounded
function with countably many discontinuities may not be Riemann
integrable. (Simple example: the characteristic function of the
rationals.)

 Quote: 2. "A necessary and sufficient condition for a function f to be Riemann integrable on a finite closed interval [a, b] is that f is bounded on [a, b] and almost everywhere continuous on [a, b]". In this theorem, can "f is bounded on [a, b]" be changed into "f is almost everywhere bounded on [a, b]"?

No. The first prerequisite for talking about Riemann integrability of
a function is to deal with the sup and the inf of the function on
subintervals determined by a partition, and you can't do that if
they're infinite. (Well, you CAN, but the Riemann sums will be
infinite, or undefined.) Defining the Riemann integral as the limit of
a net of Riemann sums only disguises the difficulty, it doesn't remove
it. (The two definitions are equivalent.)

If a function is unbounded on an interval, it CANNOT be Riemann
integrable there. (It can have an improper Riemann integral, but
that's an unhappy term, one likely to confuse students.)

 Quote: 3. "If f is continuously differentiable on a finite interval..." Here: "continuously differententiable" means the function itself is continuous and differentiable or the function's derivative is continuous?

The latter. The former would be superfluous, since differentiability
IMPLIES continuity. Mathematicians, unlike politicians, eschew
prolixity.

 Quote: 4. "A finite variation function on [a, b] is differentiable almost everywhere on [a, b]"... 5. This is my own statement: "monotonic functions always have left and right derivatives, but their left and right derivatives are not neccessarily equal." Is this statement correct?

Most emphatically not. What are the left and right derivatives of the
cube root function, at x = 0?

You are probably confusing this with the (true) result that a monotonic
function always has left- and right-sided limits at any point. That's
trivial. The result about differentiability is far from trivial.

 Quote: How to prove this? -------------------------------------- The above are just examples of questions I had in mind while reading a stochastic calculus book. Since this book is not on analysis per se, it does not give detailed account about the proof and logics behind statements... As you can see I really need "reference books" that can be put on my bookshelf that I can referred to from time to time.

One of the most popular analysis books is by Folland. I actually don't
like Folland; I think it's poorly written. "Measure and Integration"
by Munroe contains the basics, if you can still find it. And for a
concrete, hands-on treatment of differentiability vs. monotonicity,
it's hard to beat Riesz-Nagy. (I think this is reprinted, in
paperback.)

Rudin's "Principles of Mathematical Analysis" does Riemann integration
very well, but contains nothing about differentiability of monotone
functions. Hewitt and Stromberg is also very good. There are dozens
and dozens of books out there.

--
Ron Bruck
Dave L. Renfro
science forum Guru

Joined: 29 Apr 2005
Posts: 570

Posted: Tue Jul 18, 2006 8:54 pm    Post subject: Re: please recommend good reference book on mathematics?

Ronald Bruck wrote (in part):

 Quote: No, a bounded function with countably many discontinuities may not be Riemann integrable. (Simple example: the characteristic function of the rationals.)

Uhh, you might want look back over this again!

I suspect you were thinking of the property of being
able to re-define a function on a countable set in order
to produce a continuous function (i.e. the function
agrees with some continuous function on a co-countable
set).

Dave L. Renfro
porky_pig_jr@my-deja.com1
science forum Guru Wannabe

Joined: 08 May 2005
Posts: 102

Posted: Tue Jul 18, 2006 9:03 pm    Post subject: Re: please recommend good reference book on mathematics?

Most of the questions you ask are from analysis (intro level) which I'm
currently studying.
I'm not sure if 'finite variations' is a part of analysis. Bounded
variations certainly is a part. I"m not familiar with anything like
'analysis reference', though. There are some good books on analysis,
I've found Kirkwood very nice, on introductory level, and Apostol - on
more advanced level. Probably you can get either textbook, browse
through them, get the answers of most of your questions. You can also
search the web. Planetmath is a good source, and wikipedia (even if
it's frown upon by many at sci.math) still contains some good articles.

 Quote: 1. In a text book, "Riemann integral can be defined for continuous functions and can be extended to functions with finitely many discontinuities". I am not sure if this "finitely many" means finite, or countable?

Well, 'finitely many' means 'finitely many' whereas 'countable' means
either finitely many or infinitely countably many', but ...
 Quote: Riemann integral should be okay for functions with countably many discontinuities, right?

yes, it can be extended to the set of countable discontinuities.
Kirkwood has a nice discussion, on introductory level. In the most
general case, Riemann-Lebesgue theorem states that the function is
Riemann-integrable if and only if the set of discontinuities has
measure 0. All countable sets have measure 0 (roughly, you can't
construct any continuous interval with non-zero measure out of any
countable set).

 Quote: 2. "A necessary and sufficient condition for a function f to be Riemann integrable on a finite closed interval [a, b] is that f is bounded on [a, b] and almost everywhere continuous on [a, b]". In this theorem, can "f is bounded on [a, b]" be changed into "f is almost everywhere bounded on [a, b]"?

I don't see how it's going to work if the function is not bounded at
least at one point at [a,b]. To show that the function is
Riemann-integrable, we form the upper and lower Darboux sums (for some
reason the analysis books I've used prefer Darboux sums rather than
Riemann partitions) and the infimum of uppers should be equal the
supremum of lowers. Say, a function is not bounded at least at one
point, say, it increases without a bound. Then all upper sums have the
infinite value, so their infimum is also infinite, whereas the supremum
of the lower sums is finite.

We can deal with unbounded functions or unbounded intervals of
intergration by considering both cases as the limit of integration of
bounded functions or bounded intervals, assuming the limit exist (so
called 'improper integrals'). Still the function if Riemann integrable
is it is *bounded and continuous almost everywhere* ('almost
everywhere" = "the set of discontinuities has a measure zero").

 Quote: 3. "If f is continuously differentiable on a finite interval..." Here: "continuously differententiable" means the function itself is continuous and differentiable or the function's derivative is continuous?

The function's derivative is continuous.
Dave Seaman
science forum Guru

Joined: 24 Mar 2005
Posts: 527

Posted: Tue Jul 18, 2006 10:41 pm    Post subject: Re: please recommend good reference book on mathematics?

On Tue, 18 Jul 2006 13:41:44 -0700, Ronald Bruck wrote:
 Quote: In article , Michael michael.monkey.in.the.jungle@gmail.com> wrote: HI all, I often met with confusion about some theorems, concepts, definitions, etc. that I really need a reference book on my bookshelf that I can refer to from time to time. For example, in reading a text I came cross the following questions recently: 1. In a text book, "Riemann integral can be defined for continuous functions and can be extended to functions with finitely many discontinuities". I am not sure if this "finitely many" means finite, or countable? Riemann integral should be okay for functions with countably many discontinuities, right? It says finite, not countable. And it MEANS finite. No, a bounded function with countably many discontinuities may not be Riemann integrable. (Simple example: the characteristic function of the rationals.)

That function has uncountably many discontinuities. It's discontinuous
everywhere.

A bounded function with countably many discontinuities is continuous a.e.
and therefore Riemann integrable.

--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
Ronald Bruck
science forum Guru

Joined: 05 Jun 2005
Posts: 356

Posted: Wed Jul 19, 2006 8:19 pm    Post subject: Re: please recommend good reference book on mathematics?

In article <e9jo20\$aml\$1@mailhub227.itcs.purdue.edu>, Dave Seaman
<dseaman@no.such.host> wrote:

 Quote: On Tue, 18 Jul 2006 13:41:44 -0700, Ronald Bruck wrote: In article , Michael michael.monkey.in.the.jungle@gmail.com> wrote: HI all, I often met with confusion about some theorems, concepts, definitions, etc. that I really need a reference book on my bookshelf that I can refer to from time to time. For example, in reading a text I came cross the following questions recently: 1. In a text book, "Riemann integral can be defined for continuous functions and can be extended to functions with finitely many discontinuities". I am not sure if this "finitely many" means finite, or countable? Riemann integral should be okay for functions with countably many discontinuities, right? It says finite, not countable. And it MEANS finite. No, a bounded function with countably many discontinuities may not be Riemann integrable. (Simple example: the characteristic function of the rationals.) That function has uncountably many discontinuities. It's discontinuous everywhere. A bounded function with countably many discontinuities is continuous a.e. and therefore Riemann integrable.

Oops! Mea culpa. Dunno WHAT I was thinking of...

--
Ron Bruck
Michael11
science forum Guru Wannabe

Joined: 15 Aug 2005
Posts: 103

Posted: Wed Jul 19, 2006 11:45 pm    Post subject: Re: please recommend good reference book on mathematics?

"Ronald Bruck" <bruck@math.usc.edu> wrote in message
news:180720061341443163%bruck@math.usc.edu...
 Quote: In article , Michael michael.monkey.in.the.jungle@gmail.com> wrote: HI all, I often met with confusion about some theorems, concepts, definitions, etc. that I really need a reference book on my bookshelf that I can refer to from time to time. For example, in reading a text I came cross the following questions recently: 1. In a text book, "Riemann integral can be defined for continuous functions and can be extended to functions with finitely many discontinuities". I am not sure if this "finitely many" means finite, or countable? Riemann integral should be okay for functions with countably many discontinuities, right? It says finite, not countable. And it MEANS finite. No, a bounded function with countably many discontinuities may not be Riemann integrable. (Simple example: the characteristic function of the rationals.) 2. "A necessary and sufficient condition for a function f to be Riemann integrable on a finite closed interval [a, b] is that f is bounded on [a, b] and almost everywhere continuous on [a, b]". In this theorem, can "f is bounded on [a, b]" be changed into "f is almost everywhere bounded on [a, b]"? No. The first prerequisite for talking about Riemann integrability of a function is to deal with the sup and the inf of the function on subintervals determined by a partition, and you can't do that if they're infinite. (Well, you CAN, but the Riemann sums will be infinite, or undefined.) Defining the Riemann integral as the limit of a net of Riemann sums only disguises the difficulty, it doesn't remove it. (The two definitions are equivalent.) If a function is unbounded on an interval, it CANNOT be Riemann integrable there. (It can have an improper Riemann integral, but that's an unhappy term, one likely to confuse students.) 3. "If f is continuously differentiable on a finite interval..." Here: "continuously differententiable" means the function itself is continuous and differentiable or the function's derivative is continuous? The latter. The former would be superfluous, since differentiability IMPLIES continuity. Mathematicians, unlike politicians, eschew prolixity. 4. "A finite variation function on [a, b] is differentiable almost everywhere on [a, b]"... 5. This is my own statement: "monotonic functions always have left and right derivatives, but their left and right derivatives are not neccessarily equal." Is this statement correct? Most emphatically not. What are the left and right derivatives of the cube root function, at x = 0? You are probably confusing this with the (true) result that a monotonic function always has left- and right-sided limits at any point. That's trivial. The result about differentiability is far from trivial. How to prove this? -------------------------------------- The above are just examples of questions I had in mind while reading a stochastic calculus book. Since this book is not on analysis per se, it does not give detailed account about the proof and logics behind statements... As you can see I really need "reference books" that can be put on my bookshelf that I can referred to from time to time. One of the most popular analysis books is by Folland. I actually don't like Folland; I think it's poorly written. "Measure and Integration" by Munroe contains the basics, if you can still find it. And for a concrete, hands-on treatment of differentiability vs. monotonicity, it's hard to beat Riesz-Nagy. (I think this is reprinted, in paperback.) Rudin's "Principles of Mathematical Analysis" does Riemann integration very well, but contains nothing about differentiability of monotone functions. Hewitt and Stromberg is also very good. There are dozens and dozens of books out there. -- Ron Bruck

Hi Ron,

Thanks a lot for your answer. It will take some time for me to digest your
post.

For the purchasing of reference book, I really like to limit the scope to a
small number of "encyclopedia" type reference books, because as a student, I
don't have much extra cash around. And also I am not specialized in the
analysis field per se, I am doing stochastic/probability stuff more on the
applied and engineering side. Perhaps I won't afford to have tens of books
solely on analysis on my bookshelf.

Also books give trouble when it comes to moving.

Library should have been a good source but I still like some essential books
to be handy.

There are two types of good books -- good text books and good
encyclopedia/reference books.

I know Rudin's and others you've pointed to me are excellent textbooks, but
are they good encyclopedia/reference books?

The "monotonicity" questions I post previously were just one samples of my
questions, I will have other questions such as "interchange of expectation
and sum", etc. I hope to find/buy a few essential books that can cover
most-used analysis concepts, theorems, etc.

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