Some Results on the Variation of Composite Function of Functions of Bounded d Variation

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1 AMRICAN JOURNAL OF UNDRGRADUAT RSARCH OL. 6 NO. 4 Some Results o the ritio o Composite Fuctio o Fuctios o Boue ritio Mohse Soltir Deprtmet o Mthemtics Fculty o Sciece K.N. Toosi Uiversity o Techology P.O. Box Tehr IRAN Receive: Februry Accepte: December ABSTRACT I this pper we iscuss coitios uer which the composite uctio o two uctios o boue vritio is lso o boue vritio. I. INTRODUCTION Cmille Jor (838-9 itrouce the clss o ll uctios o boue vritio clsse them s iereces o o-ecresig uctios. These uctios ply cetrl role i my ivestigtios otbly i stuies o rectiibility i stuies ito umetl questios ivolvig itegrls erivtives. They lso le to turl geerliztios i the bstrct stuies o mesure itegrtio. Toy my text boos i rel lysis cover them i reltio to the Riem-Stieltjes itegrbility o rel vlue uctio eie o the close itervl [. [-3] By cosierig these uctios we see tht or the uclie metric the boue vritio is ivrit uer rithmetic opertios + - ; tht is the rithmetic opertios + - with two uctios g o boue vritio o [ with respect to the uclie metric give uctios ± g g o boue vritio o [ with respect to the sme metric. (N.B. The boue vritio is ot ivrit uer the rithmetic opertio becuse i the Cotct: soltir@si.tu.c.ir ivisio cse the eomitor uctio my ot be r rom zero; tht is 0 ( Im g cosequetly the uctio g is uboue o [ so is ot boue vritio o [. As we hve see beore the results o compositio re lwys stuie iscusse log wit the results o rithmetic opertios + - i subjects lie cotiuity ieretibility itegrbility [4]. I this pper Chistyo s 997 eiitio o boue vritio is use to el with the possible results o vritio o composite uctio go o two uctios o boue vritio g eie o [ [c] respectively with Im( [ c ]. I prticulr or the cse the results the bove metioe results o rithmetic opertios + - me cosistet complete iscussio o ll primry opertios o uctios i the subject o uctios o boue vritio versus the cosistet complete iscussio o ll primry opertios o uctio o cotiuity ieretibility itegrbility o uctios. For the uclie metric cse it is showe tht espite the rithmetic opertios + - similr to the 5

2 AMRICAN JOURNAL OF UNDRGRADUAT RSARCH OL. 6 NO. 4 rithmetic opertio boue vritio is ot ivrit uer the compositio opertio. I the irst two theorems some coitios o tow uctios o boue vritio re represete tht imply their composite uctio is lso o boue vritio. By cosierig geerl metric iste o i the thir theorem ecessry o-suiciet coitio o the metric is stte to hve the property o the ivrice o boue vritio uer the compositio opertio. The lst theorem els with etermiig ctegory o metrics tht hve the property o the ivrice o boue vritio uer compositio. II. PRLIMINARIS The reer who hs stuie the cocept o uctios o boue vritio shoul be well cquite with the ollowig eiitios results. For essetil ccout o the theory o uctio o boue vritio with respect to metric see reerece [5]. I the iscussio throughout this pper it is ssume tht the metric is cosiere s rel-vlue uctio rom the -imesiol uclie spce (R ito the -imesiol uclie spce (R. Furthermore it is te or grte tht both uctios g re eie o boue close itervls i the rel lie. For the Mi Results some ottios eiitios re eee. We eote the set o ll Prtitios o [ by P. Tht is b P N < < K < < }... b P } [ { 0 : 0 b Let be uctio eie o [. For ech prtitio P { o [ put } 0 ( P ( ( (. The umber ( P is clle the vritio o o [ or prtitio P. Deiitio. The uctio is si to be o boue vritio o [ provie the set { ( P : P P b } is boue bove. I is uctio o boue vritio the we set ( [ sup{ ( P : P P b }. We cll ( [ the vritio o o [ with respect to metric. Deiitio. The metric is si to hve the property o the Ivrice o Boue ritio uer Compositio Opertio (IBCO property i or y two uctio g o boue vritio o [ [c] respectively with Im( [ c ] go Is uctio o boue vritio o [. The ollowig result rom propositio.4.3 o [6] (pges will be useul i the rest o this pper. Propositio. Let be uctio eie o [ re two Lipschitz equivlet metrics. The is boue vritio o [ i oly i it is boue vritio o [. The ollowig corollry is strightorwr geerliztio o Theorem o reerece [] (pge 3. Corollry. Suppose tht is uctio o 0 boue vritio o [ { } is prtitio o [. The is uctio o boue vritio o ech [ - ] urthermore: III. ( [ ] ( [ ( ( [ ] ( [ MAIN RSULTS or. ( I this sectio we ite to show tht the metric plys importt role i boueess o totl vritio o composite uctios. At this stge we metio tht ll theorems below with respect to metric re 6

3 AMRICAN JOURNAL OF UNDRGRADUAT RSARCH OL. 6 NO. 4 vli with little chges or ll Lipschitz equivlet metrics with (see Propositio. We begi with the rther cocrete cse. To hve similr theorem to the theorems o rithmetic opertios + - we turlly expect to give irmtive swer to the ollowig questio. Questio. Does the metric hve the IBCO property? O the other h the couter exmples below give egtive swer to the questio. xmple. Cosier uctios o boue vritio lie g eie o [0] [-] respectively by: ( x x si( π x i (x 0 (0 0 g(x i (x 0 g(0 0. The the composite uctio g o is eie o [0] by: go (x 0 i (x 0 with 3 go (x i (x 0 with 3. Put + K + { ( ( } P 0 where 3 The P is prtitio o [0] ( g o P ( where 3 Accorigly ( g o [ 0]. xmple. Cosier uctios o boue vritio lie g eie o [0] respectively by: ( x i ( x 0 i ( x K ( x K ( g x x. The the composite uctio g o is eie o [0] by: g o ( x i ( x 3K i ( x 3K g o (x 0. Put P { 0 ( 0 + ( + ( + } K 3 where 3 The is prtitio o [0] P ( g P o where 3 Accorigly ( g [ 0] o. The couterexmples bove show tht the compositio o two uctios o boue vritio is ot ecessrily itsel o boue vritio. Nevertheless there exist remrble specil cses i the composite uctio o two uctio o boue vritio is o boue vritio too. As the irst cse we suppose tht g is rbitrry uctio o boue vritio o [c]. We s i there re suiciet coitios o the uctio o vritio o [ with Im( [ c ] implyig the boue vritio o the composite uctio g o o [. By observig couple o couterexmples we c see tht the uctio i both couterexmples hs commo property 0 tht is there is ot y prtitio { } o [ which is mootoe o ech [ ] or. I we elimite this commo property the we re le to the ollowig theorem. Theorem. Let be rbitrry uctio o boue vritio o [ let 0 there exist prtitio { } o [ which is mootoe o ech [ ] or. The or y uctio g o boue vritio o [c] with Im( [ c ] g o is boue vritio o [ urthermore: ( g [ g [ c ] o (. (3 Proo. Let suppose tht is icresig o [ ] (the proo is logous i the other cse. Let 7

4 AMRICAN JOURNAL OF UNDRGRADUAT RSARCH OL. 6 NO. 4 P { } m l be prtitio o [ ] l 0 P { ( } m l l 0 [ ( ] ( so is prtitio o (see qutio o Corollry : ( g [ ( ( ] ( g [ c ] (4 We lso hve: m ( g o P ( g o ( l g o ( l l m l ( g( ( g( ( ( g P. l l Hece we c write: Now equtios (4 (5 give ( g [ o sup { ( g o P P P } ] : ( sup { ( g P : P P } ( g [ ( ( ] (. (5 ( g [ ] ( g [ c ] o or. Tig summtio o these iequlities implies: ( g o [ ] ( g [ c. ] Thereore by cosierig equtio ( o Corollry the esire sttemet is prove.. Corollry. Let ( x x be polyomil with rel coeiciets eie o [. I g is uctio o boue vritio o [c] such tht Im( [ c ] the g o is uctio o boue vritio o [. Now s the seco cse we cosier g iste o. Suppose tht is rbitrry uctio o boue vritio o [. We s i there re suiciet coitios o the uctio g o boue vritio o [c] with Im( [ c ] implyig the boue vritio o the composite uctio g o o [. O pge 406 o reerece [7] it hs bee prove tht every uctio g o boue vritio is lmost everywhere ieretible. I g is ot ieretible t eve oe poit o (c the xmple shows tht g is ipproprite choice or our purpose so we hve to the limittio o ieretibility o g o (c. 8

5 AMRICAN JOURNAL OF UNDRGRADUAT RSARCH OL. 6 NO. 4 Furthermore i g is ieretible o (c but g' is uboue the xmple shows tht g is lso ipproprite choice or our purpose. I we the boueess o g' o (c s the seco limittio the we i the pproprite choice this les us to the ext theorem. Theorem. Let be rbitrry uctio o boue vritio o [. The or y uctio g o boue vritio o [c] with boue erivtive g' o (c Im( [c ] g o is uctio o boue vritio o [ urthermore: ( g o [ 4( sup g( x ( sup g'( x ( [. c x + c < x < Proo. By mes o Corollry 4 o reerece 7 (pge 4 or every x y (c: ( g( x g( y ( sup g'( x ( x y c < x < Now the bove iequlity the rbitrrilygive prtitio o [ give: { } P 0 ( g o P ( g o ( g o ( ( g o ( g( 0 + ( g o ( g o ( + ( g o ( g o ( ( ( g( ( g( ( + g( ( g( ( ( g'( x ( P. < 4 sup c < x < Thereore we c coclue tht: 0 + ( g( ( g( ( ( g [ sup{ ( g o P P o P } : ( sup g( x ( sup g'( x ( [ ]. 4 b c x + c < x < b Corollry 3. Let be uctio o boue vritio o [ the exp ( is uctio o boue vritio o the sme itervl. Corollry 4. Suppose uctio positive r rom zero uctio g re both boue vritio o [. The the uctio o [. g is lso o boue vritio Proo. By mes o Theorem l( is uctio o boue vritio so is. Now Corollry 3 implies the g l ( ( ( boue vritio oexp g l g or. 9

6 AMRICAN JOURNAL OF UNDRGRADUAT RSARCH OL. 6 NO. 4 I sum the bove theorems results show tht the uclie metric oes ot hve the IBCO property. The ollowig theorem gives chrcteriztio o the metric which hs the IBCO property lso s corollry it shows tht the metric oes ot hve the ecessry coitio o hvig tht property. Theorem 3. Let the metric hve the IBCO property. The the metric s rel vlue uctio :( R (R is iscotiuous. Proo. Assume to rech cotrictio tht the metric is cotiuous. Cosier the icresig sequece { } y or ll (( y ( 0 y where Ν i [0]. Sice lim so by our recet ssumptio lim (( y ( 0 The there exists sub-sequece { } y such tht ( y < or ll Ν. This implies tht <. Now i we eie uctio o [0] by: ( x i (x y ( x i (x y y ( y the x y ( [0] sup ( x y + ( y < 0 Furthermore we cosier g o [0] eie by: g( x 0 i (x g( x i (x. The ( g [0] (0 <. Filly g o is eie o [0] by: g o ( x i ( x y ( x y... g o ( x 0 i... which implies g o [0]. Thus by Deiitio ( we hve cotrictio to the ssumptio tht the metric hs the IBCO property. As the irst remr we poit out tht Theorem 3 gives merely ecessry coitio o hvig the IBCO property. For exmple eie metric ( R ( R : ( x y ( h( x h( y Where the bijectio h :R R is eie by: h (x x i x (0 h(x -x + h (x -x + i x (0. The metric is iscotiuous. To prove it cosier the sequece ( 0 where { } (( 0(00 0. lim Now eie uctios g similr to xmple. I this cse ( [0] ( g [0] < <. For the composite uctio g o tig the prtitios P s were te i xmple gives: ( g P Accorigly ( g o [0]. o where 3 To i the pproprite metrics tht hve the IBCO property Theorem 3 suggests tht we shoul ocus our ttetio o ll iscotiuous metrics : ( R ( R. The ext theorem iscusses the bove metioe poit. Theorem 4. Let the metric stisy the coitio: i ( x y > 0. < x y < 0

7 AMRICAN JOURNAL OF UNDRGRADUAT RSARCH OL. 6 NO. 4 The it hs the IBCO property. Proo. First let be rbitrry uctio o boue vritio o [. We clim tht is step uctio. Suppose tht this is ot the cse so tht there exists icresig sequece { } o [ such tht x x ( x or ll Ν. Hece ( + tig the prtitios { x x... x b} o [ gives: P ( P ( ( ( x + ( ( x ( x + ( ( x ( b ( i < x y < ( x y or... So by eiitio ( [ which is cotrictio to our ssumptio. Seco sice is step uctio so or y uctio o boue vritio g o [c] with Im( [ c ] g o is lso step uctio o [. evetully every step uctio is o boue vritio o [ this completes the proo. Corollry 5. The iscrete metric 0 hs the IBCO property. As the seco remr we poit to the crility o the set o ll metrics ( ( R R tht hve the : IBCO property is equl to the crility o the set o ll metrics tht is c where c eotes the cotiuum. To Prove this or y uctio : R R with i ( x > 0 eie metric by: < x < : R ( ( R ( x y ( ( x + ( y ( x y. Now sice the crility o the set o ll uctios : R R with c i < x < ( x > 0 is or ech elemet o this set i < x y < ( x y i < x < ( x by mes o Theorem 4 the clime ssertio is prove. I the e while Theorem 3 gives ecessry coitio theorem 4 gives suiciet coitio or the metric to hve 0 the IBCO property it remis or the itereste reer to swer the ollowig: Questio. Is there y ecessry suiciet coitio o the metric to hve the IBCO property? I so wht is tht coitio? ACKNOWLDGMNTS The uthor woul lie to ths Dr. F. Mle Dr. K. Nourouzi or their time vice. Specil ths to Dr. H. Hghighi or his ivluble support. RFRNCS. H.S. Gsill P.P. Nryswmi Foutios o Alysis (Hrper & Row Publishers New Yor 989 pges M.J. Schrmm Itrouctio to Rel Alysis (Pretice-Hll glewoo Clis NJ 996 pges J.D. Depree C.W. Swrtz Itrouctio to Rel Alysis (Wiley New Yor 988 pges W. Rui Priciples o Mthemticl Alysis (McGrw Hill New Yor 988 pges Chistyo O Mppig o Boue ritio J. Dym. Cotrol Systems ol. 3 No. (997 pges W.A. Sutherl Itrouctio to Metric Topologicl Spces (Clreo press Oxor C.C. Pugh Rel Mthemticl Alysis (Spriger-erlg NewYor 00.

8 AMRICAN JOURNAL OF UNDRGRADUAT RSARCH OL. 6 NO. 4

Review of the Riemann Integral

Review of the Riemann Integral Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of