Lipschitz Functions and Riemann Integral
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From: Rob Johnson on 21 May 2010 17:22 In article <210520101142328320%bruck(a)math.usc.edu>, Ronald Bruck <bruck(a)math.usc.edu> wrote: >In article ><2000510524.174686.1274111768185.JavaMail.root(a)gallium.mathforum.org>, >Maury Barbato <mauriziobarbato(a)aruba.it> wrote: > >> Hello, >> let R be the field of real numbers. >> Does there exist some Lipschitz function g:[a,b]->[c,d] >> and some bounded function f:[c,d]-> R wich is Riemann >> integrable such that f°g is not Riemann integrable? >> >> Thank you very much for your attention. >> My Best Regards, >> Maury Barbato >> >> PS If one only supposes that g is continuous, the answer >> is yes. Let f be the characteristic function of the >> ternary Cantor set C, and g a homeomorphism from a "fat" >> Cantor set onto C. Then f is Riemann integrable on [0,1], >> but f°g is not Riemann integrable. > >I have no idea what this symbol is supposed to be. On my Mac, it looks >like an asterisk, but when I save it to file and examine it with od I >find it actually reads > > f "0xc2 0xb0" g > >whatever that's supposed to be. (Composition?) > >For not-the-first, and almost certainly not-the-last, time, when you >post to Usenet, USE ONLY STRAIGHT ASCII. That will bring all witless >OS's, such as Windows, back to reality. (Actually, Windows is >perfectly cable of handling Unicode, but nobody's programs, including >Microsoft's, seem to utilize it. Perhaps this is Unicode, and my Mac >can't handle it. But it's supposed to, out-of-the-box.) I use Thunderbird 3.0.4 on OS X 10.5.8, and I see the character as the degree symbol (probably intended as functional composition). The header of Maury's post says >Content-Type: text/plain; charset=UTF-8 >Content-Transfer-Encoding: 8bit and the two byte sequence C2B0 is the UTF-8 encoding for the Unicode character 00B0, which is "DEGREE SIGN". This character doesn't have a standard 7-bit ASCII encoding, however, I had the impression that Thoth 1.8.3 understood UTF-8. What version of OS X are you running? Rob Johnson <rob(a)trash.whim.org> take out the trash before replying to view any ASCII art, display article in a monospaced font
From: Rob Johnson on 21 May 2010 17:30 In article <20100521.120653v3(a)whim.org>, Rob Johnson <rob(a)trash.whim.org> wrote: >In article <210520101142328320%bruck(a)math.usc.edu>, >Ronald Bruck <bruck(a)math.usc.edu> wrote: >>In article >><2000510524.174686.1274111768185.JavaMail.root(a)gallium.mathforum.org>, >>Maury Barbato <mauriziobarbato(a)aruba.it> wrote: >> >>> Hello, >>> let R be the field of real numbers. >>> Does there exist some Lipschitz function g:[a,b]->[c,d] >>> and some bounded function f:[c,d]-> R wich is Riemann >>> integrable such that f°g is not Riemann integrable? >>> >>> Thank you very much for your attention. >>> My Best Regards, >>> Maury Barbato >>> >>> PS If one only supposes that g is continuous, the answer >>> is yes. Let f be the characteristic function of the >>> ternary Cantor set C, and g a homeomorphism from a "fat" >>> Cantor set onto C. Then f is Riemann integrable on [0,1], >>> but f°g is not Riemann integrable. >> >>I have no idea what this symbol is supposed to be. On my Mac, it looks >>like an asterisk, but when I save it to file and examine it with od I >>find it actually reads >> >> f "0xc2 0xb0" g >> >>whatever that's supposed to be. (Composition?) >> >>For not-the-first, and almost certainly not-the-last, time, when you >>post to Usenet, USE ONLY STRAIGHT ASCII. That will bring all witless >>OS's, such as Windows, back to reality. (Actually, Windows is >>perfectly cable of handling Unicode, but nobody's programs, including >>Microsoft's, seem to utilize it. Perhaps this is Unicode, and my Mac >>can't handle it. But it's supposed to, out-of-the-box.) > >I use Thunderbird 3.0.4 on OS X 10.5.8, and I see the character as >the degree symbol (probably intended as functional composition). The >header of Maury's post says > >>Content-Type: text/plain; charset=UTF-8 >>Content-Transfer-Encoding: 8bit > >and the two byte sequence C2B0 is the UTF-8 encoding for the Unicode >character 00B0, which is "DEGREE SIGN". This character doesn't have >a standard 7-bit ASCII encoding, however, I had the impression that >Thoth 1.8.3 understood UTF-8. What version of OS X are you running? I am sorry about the multiple posts. I was trying to get my system to post in UTF-8. Finally I got it to work after three posts. Rob Johnson <rob(a)trash.whim.org> take out the trash before replying to view any ASCII art, display article in a monospaced font
From: cwldoc on 21 May 2010 15:34
> In article > <2000510524.174686.1274111768185.JavaMail.root(a)gallium > .mathforum.org>, > Maury Barbato <mauriziobarbato(a)aruba.it> wrote: > > > Hello, > > let R be the field of real numbers. > > Does there exist some Lipschitz function > g:[a,b]->[c,d] > > and some bounded function f:[c,d]-> R wich is > Riemann > > integrable such that f°g is not Riemann integrable? > > > > Thank you very much for your attention. > > My Best Regards, > > Maury Barbato > > > > PS If one only supposes that g is continuous, the > answer > > is yes. Let f be the characteristic function of the > > ternary Cantor set C, and g a homeomorphism from a > "fat" > > Cantor set onto C. Then f is Riemann integrable on > [0,1], > > but f°g is not Riemann integrable. > > I have no idea what this symbol is supposed to be. > On my Mac, it looks > like an asterisk, but when I save it to file and > examine it with od I > find it actually reads > > f "0xc2 0xb0" g > > whatever that's supposed to be. (Composition?) > > For not-the-first, and almost certainly not-the-last, > time, when you > post to Usenet, USE ONLY STRAIGHT ASCII. That will > bring all witless > OS's, such as Windows, back to reality. (Actually, > Windows is > perfectly cable of handling Unicode, but nobody's > programs, including > Microsoft's, seem to utilize it. Perhaps this is > Unicode, and my Mac > can't handle it. But it's supposed to, > out-of-the-box.) > > --Ron Bruck Not that anyone cares, but the symbol in question is the "degree" symbol in unicode. I do not know if there is a symbol in unicode for composition of functions. Actually, there is a symbol in unicode called a "composition symbol," but it does not look like anything I've ever seen before. The code for the degree symbol is 176 (b0 in hexadecimal). Because the code is greater than 127 and less than 192, it ends up being stored in the form of two bytes in UTF-8 (stands for Unicode Transformation Format, using 8-bit blocks to represent a character) encoding. The 8-bit symbol code, 1011 0000, is stored across two bytes as follows. The bytes have the values 110x xxxx and 10xx xxxx, where the bits of the symbol code then get substiuted for the x's in the final bytes. So this ends up being 1100 0010 and 1011 0000, which is c2 b0. |