From mboxrd@z Thu Jan 1 00:00:00 1970 X-Spam-Checker-Version: SpamAssassin 3.4.4 (2020-01-24) on polar.synack.me X-Spam-Level: X-Spam-Status: No, score=-1.3 required=5.0 tests=BAYES_00,INVALID_MSGID autolearn=no autolearn_force=no version=3.4.4 X-Google-Language: ENGLISH,ASCII-7-bit X-Google-Thread: 109fba,baaf5f793d03d420 X-Google-Attributes: gid109fba,public X-Google-Thread: fc89c,97188312486d4578 X-Google-Attributes: gidfc89c,public X-Google-Thread: 1014db,6154de2e240de72a X-Google-Attributes: gid1014db,public X-Google-Thread: 103376,97188312486d4578 X-Google-Attributes: gid103376,public From: thp@cs.ucr.edu (Tom Payne) Subject: Re: What's the best language to start with? [was: Re: Should I learn C or Pascal?] Date: 1996/09/27 Message-ID: <52gvu3$jhb@rumors.ucr.edu>#1/1 X-Deja-AN: 185688438 references: <01bb8df1$2e19d420$87ee6fce@timpent.airshields.com> <322f864d.42836625@news.demon.co.uk> <01bb9bf9$61e9e0e0$87ee6fce@timpent.airshields.com> <50sj6q$aci@mtinsc01-mgt.ops.worldnet.att.net> <01bb9d25$9cb3cb00$32ee6fcf@timhome2> <50v6k3$soo@mtinsc01-mgt.ops.worldnet.att.net> <01bb9ded$cd0fdf00$32ee6fcf@timhome2> <5136on$7qj@goanna.cs.rmit.edu.au> <01bb9f26$36c870e0$87ee6fce@timpent.a-sis.com> <1996Sep24.133312.9745@ocsystems.com> organization: University of California, Riverside newsgroups: comp.lang.c,comp.lang.c++,comp.unix.programmer,comp.lang.ada Date: 1996-09-27T00:00:00+00:00 List-Id: In comp.lang.c++ Joel VanLaven wrote: [...] : Actually, not all functions are integrable. The most complete and : irrefutable definition of the integral of a function to my knowledge is : : Given a partition P of [a,b] : (P is a finite subset of [a,b] including a and b) : P={x0,x1,x2,...xn} such that a=x0, b=xn, and x(j+1)>xj : call Mj the lub(f([x(j-1),xj]) (least upper bound) : call mj the glb(f([x(j-1),xj]) (greatest lower bound) : The number Uf(P)=SUM(Mj(xj-x(j-1))) 1<=j<=n : is called the P upper sum for f : The number Lf(P)=SUM(mj(xj-x(j-1))) 1<=j<=n : is called the P lower sum for f : The unique number I that satisfies the inequality ^^^^^^^^^^^^^^^^^^^ if it exists : Lf(P)<=I<=Uf(P) for all possible P of [a,b] : is called the definite integral of f from a to b. Actually, there is a significant generalization, called Lebesgue integration, that can be found in any graduate text on Real Analysis (e.g., those by Royden and by Rudin). Nevertheless, if you believe in the axiom of choice, there are still functions that are not integrable (but they get pretty weird). Tom Payne