# S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Thu, 20 Jun 2013 04:19:45 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 5 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: True or FalsePosted: Mon, 16 Apr 2012 03:21:12 UTC
 Senior Member

Joined: Sat, 10 Oct 2009 15:43:22 UTC
Posts: 98
If f is continuous on (a,b), the f(a) and f(b) can be defined so that f is integrable on [a,b].

My inclination is to say false and to use sin(1/x) as a counterexample. However, since this is a problem I would need to present on the board using "because" as my explanation probably wouldn't be sufficient. How might I explain this accurately and concisely? Any help appreciated!

MF

_________________
let:
a=b
ab=b^2
ab-a^2=b^2-a^2
a(b-a)=(b+a)(b-a)
a=b+a
a=2a
1=2

Top

 Post subject: Re: True or FalsePosted: Mon, 16 Apr 2012 04:29:27 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6068
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
mgfortes wrote:
If f is continuous on (a,b), the f(a) and f(b) can be defined so that f is integrable on [a,b].

My inclination is to say false and to use sin(1/x) as a counterexample. However, since this is a problem I would need to present on the board using "because" as my explanation probably wouldn't be sufficient. How might I explain this accurately and concisely? Any help appreciated!

MF

I assume you are dealing with Riemann integral rather than the many other definitions of integral?

Recall Lebesgue's characterisation of Riemann integrable functions. {a,b} is null in [a,b], so it doesn't matter what you define f(a), f(b) to be.

_________________

Top

 Post subject: Re: True or FalsePosted: Mon, 16 Apr 2012 06:02:55 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12172
Location: Austin, TX
outermeasure wrote:
mgfortes wrote:
If f is continuous on (a,b), the f(a) and f(b) can be defined so that f is integrable on [a,b].

My inclination is to say false and to use sin(1/x) as a counterexample. However, since this is a problem I would need to present on the board using "because" as my explanation probably wouldn't be sufficient. How might I explain this accurately and concisely? Any help appreciated!

MF

I assume you are dealing with Riemann integral rather than the many other definitions of integral?

Recall Lebesgue's characterisation of Riemann integrable functions. {a,b} is null in [a,b], so it doesn't matter what you define f(a), f(b) to be.

Indeed, you can have a much simpler counter example, use on .

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Post subject: Re: True or FalsePosted: Mon, 16 Apr 2012 21:45:30 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Wed, 21 May 2003 04:27:18 UTC
Posts: 996
mgfortes wrote:
If f is continuous on (a,b), the f(a) and f(b) can be defined so that f is integrable on [a,b].

My inclination is to say false and to use sin(1/x) as a counterexample. However, since this is a problem I would need to present on the board using "because" as my explanation probably wouldn't be sufficient. How might I explain this accurately and concisely? Any help appreciated!

MF

Everything said above is certainly true...on the other hand, you might be referring to uniform continuity, since we can say something like this:

A function f is uniformly continuous on the interval (a,b) if and only if it can be defined at the endpoints a and b such that the extended function is continuous on [a,b].

This is sometimes called the "Continuous Extension Theorem". The essential part of the argument to verify it is that if we assume f to be uniformly continuous and the sequence {x_k} is Cauchy, then the image sequence {f(x_k)} is also Cauchy.

Therefore, you may choose to rephrase the original question as follows:

Quote:
If f is continuous on (a,b), then f(a) and f(b) can be defined so that f is integrable on [a,b] (trivally true, see Outermeasure's post) AND will be (properly) Riemann integral (not necessarily true, see Shadow's example).

_________________
"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

Top

 Post subject: Re: True or FalsePosted: Tue, 17 Apr 2012 10:33:33 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6068
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Indeed, you can have a much simpler counter example, use on .

Indeed, and actually sin(1/x) is not a counterexample since it is bounded near 0.

_________________

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 5 posts ]

 All times are UTC [ DST ]

#### Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous