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Is there an example of a discontinuous integrable f and a continuous integrable g such that f(g(x)) is non-integrable? I know the converse is not true - if f is continuous and integrable, and g is integrable, then f(g(x)) is integrable, but what about with g continuous? I'm fairly confident there is no counterexample, but I'm not totally sure how to prove it - how would I go about it if that is the case?
Thanks very much,
Mathmos6 ( talk) 13:38, 15 April 2009 (UTC)
Well the standard example for f-integrable g-integrable with composition fg not integrable is g Thomae's function, f(x)=1 everywhere except f(0)=0 - so is the next step something similar to that? I can't honestly say I'm completely sure how to go ahead, despite the fact you've already given me a lot of help - sorry, my head's obviously having a slow night!
Mathmos6 ( talk) 19:06, 15 April 2009 (UTC)
That's brilliant! I understand completely I think, except for one thing - what's the relevance of h being homeomorphic? Is that just so we know it's continuous, or does it have additional relevance? As a matter of interest, is it safe to 'adjust' the example so that g is [0,1]->[0,1], say by simply multiplying h(x) by (1/2)? Thanks so much for the help, Mathmos6 ( talk) 04:46, 16 April 2009 (UTC)
does poles belong to an isolated singularity?? —Preceding unsigned comment added by 59.96.30.227 ( talk) 15:08, 15 April 2009 (UTC)
This is an Olympiad question which I cannot think of any way to start. "The +ve integers a and b are such that 15a + 16b and 16a - 15b are both squares of +ve integers. What is the least possible value that can be taken by the smaller of these squares." Please help. -- Siddhant ( talk) 16:54, 15 April 2009 (UTC)
How? I know the answer but I need the method. Thanks for your effort.-- Siddhant ( talk) 17:32, 15 April 2009 (UTC)
Do I need to use Fermat's theorem for that?-- Siddhant ( talk) 18:44, 15 April 2009 (UTC)
I assumed let r2=15a + 16b and let s2=16a - 15b. I squred both the squares to get r4 + s4 = 481 (a2 + b2). Now how to prove that each of the squares is divisible by 481? What arguments need to be given after that to reach the answer? Is there another approach possible to answer this question? Thanks.-- Siddhant ( talk) 08:06, 17 April 2009 (UTC)
Not the OP, but I've gotten sucked in myself. I can show that r (and/or s) is divisible by 481 (thanks to the above hints), and so get that r^2 must be a multiple of 231361, but how do I know that's actually a solution to the original equations? Don't I need to find workable values of a and b? 74.7.169.90 ( talk) 21:58, 17 April 2009 (UTC)
Mathematics desk | ||
---|---|---|
< April 14 | << Mar | April | May >> | April 16 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Is there an example of a discontinuous integrable f and a continuous integrable g such that f(g(x)) is non-integrable? I know the converse is not true - if f is continuous and integrable, and g is integrable, then f(g(x)) is integrable, but what about with g continuous? I'm fairly confident there is no counterexample, but I'm not totally sure how to prove it - how would I go about it if that is the case?
Thanks very much,
Mathmos6 ( talk) 13:38, 15 April 2009 (UTC)
Well the standard example for f-integrable g-integrable with composition fg not integrable is g Thomae's function, f(x)=1 everywhere except f(0)=0 - so is the next step something similar to that? I can't honestly say I'm completely sure how to go ahead, despite the fact you've already given me a lot of help - sorry, my head's obviously having a slow night!
Mathmos6 ( talk) 19:06, 15 April 2009 (UTC)
That's brilliant! I understand completely I think, except for one thing - what's the relevance of h being homeomorphic? Is that just so we know it's continuous, or does it have additional relevance? As a matter of interest, is it safe to 'adjust' the example so that g is [0,1]->[0,1], say by simply multiplying h(x) by (1/2)? Thanks so much for the help, Mathmos6 ( talk) 04:46, 16 April 2009 (UTC)
does poles belong to an isolated singularity?? —Preceding unsigned comment added by 59.96.30.227 ( talk) 15:08, 15 April 2009 (UTC)
This is an Olympiad question which I cannot think of any way to start. "The +ve integers a and b are such that 15a + 16b and 16a - 15b are both squares of +ve integers. What is the least possible value that can be taken by the smaller of these squares." Please help. -- Siddhant ( talk) 16:54, 15 April 2009 (UTC)
How? I know the answer but I need the method. Thanks for your effort.-- Siddhant ( talk) 17:32, 15 April 2009 (UTC)
Do I need to use Fermat's theorem for that?-- Siddhant ( talk) 18:44, 15 April 2009 (UTC)
I assumed let r2=15a + 16b and let s2=16a - 15b. I squred both the squares to get r4 + s4 = 481 (a2 + b2). Now how to prove that each of the squares is divisible by 481? What arguments need to be given after that to reach the answer? Is there another approach possible to answer this question? Thanks.-- Siddhant ( talk) 08:06, 17 April 2009 (UTC)
Not the OP, but I've gotten sucked in myself. I can show that r (and/or s) is divisible by 481 (thanks to the above hints), and so get that r^2 must be a multiple of 231361, but how do I know that's actually a solution to the original equations? Don't I need to find workable values of a and b? 74.7.169.90 ( talk) 21:58, 17 April 2009 (UTC)