Studiot,Thanks for your reply.
I am reading Calculus Vol1 of Tom.M.Apostol.There is one core question:
"How could we now whether one function is integrable"?Actually the 3 questions I asked could all integrabled into this one core question.
So how could I prove it?
I don't have access to Apostols's book, though I 've heard of it.
The 'proof' you are seeking depends upon knowing quite a bit about limits and what are known as epsilon - delta arguments and is several pages long. It is normally simply stated in elemenary treatments.
What notation does Apostol use?
Notation means the symbols used to represent something. It is easier to understand proofs and derivations if they are written in familiar symbols.
But no matter we will try to develop something.
I looked in the contents list in Apostol's book in Amazon.
He introduces partitions, upper and lower bounds and set areas before integration so presumably derives the integral as a sum 'sandwiched' between upper and lower bounds.
However he delays treatment of continuity until after integration so you must look in later chapters for some ideas.
It is not necessary for a function to be continuous to be integrable, only bounded. If a single one of the partitions is unbounded the integration fails as the sum cannot be formed.
Anyway to your proof.
You will find it (in my notation) with aome explanation in the attachment. Sorry about the handwriting, I can't get TEX to do the subscripts I want.
when you say covering the entire plane??does this mean that we are dealin with a plane or what??
plizz i can't visualise what you are talkin about!!help me!
TO prove any arbitirary function f(x) i sintegrable from inteval [a,b] you must first derive the basic calclus theorms and integration. Then you notice is a limit and a sum (assumung you derived it). If there are discontinuites, then logically the limit cannot exist HOWEVEr, the function is still integrable because you can take teh are of interval [a,M] where M is teh point right before discontinuity. If tehre are INIFINITE discontinuites then no a function is not integrable but you can only see this if you derive integration as a limit anda a sum.
since you are staring at col1, i suggest you first prove the rigrous defintion of limits (its fairly easy-my proof that is) ill post up all my mathematical proofs after i scan all of em - too lazy. Then once you understand limits, you can move onto proving diffrentiation - or the basics a tleas. and prove how to find the instantaneous slope of a some arbitraty function f(x). Then you ned to move onto integration and prving it. Again the concept is fairly simple and its REALLY cool, or atlest fo rme. Finally you can work on showing that integration is the "anti"deravitive and tehn see the strict relationship between limits and integration. hten you can conclude relationships about integration with respect to its function answering your questions. I could just state the answer - which i ckinda did- but thi sis a proccess of self-discovery