. The set of these sums. be a partition of [ ab, ]. Consider the sum f( x) f( x 1)

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1 Chapter 7 Fuctos o Bouded Varato. Subject: Real Aalyss Level: M.Sc. Source: Syed Gul Shah (Charma, Departmet o Mathematcs, US Sargodha Collected & Composed by: Atq ur Rehma (atq@mathcty.org, We shall ow dscuss the cocept o uctos o bouded varato whch s closely assocated to the cocept o mootoc uctos ad has wde applcato mathematcs. These uctos are used Rema-Steltjes tegrals ad Fourer seres. Let a ucto be deed o a terval [ ab, ] ad P= { a= x, x,..., x = b} be a partto o [ ab, ]. Cosder the sum ( x ( x. The set o these sums s te. It chages whe we make a reemet a partto. I ths set o sums s bouded above the the ucto s sad to be a bouded varato ad the supremum o the set s called the total varato o the ucto o [ ab, ], ad s ;, V ab, ad t s also alated as V( or V. deoted by V( ab or ( Thus ( = V ; ab, sup ( x ( x The supremum beg take over all the partto o [ ab, ]. Hece the ucto s sad to be o bouded varato o [ ab, ], ad oly, V ; ab, <. ts total varato s te.e. ( Note Sce or x c y, we have ( y ( x ( y ( c + ( c ( x Thereore the sum ( x ( x ca ot be decrease (t ca, act oly crease by the reemet o the partto. Theorem A bouded mootoc ucto s a ucto o bouded varato. Suppose a ucto s mootocally creasg o [ ab, ] ad P s ay partto o [ ab, ] the ( x ( x = ( ( x ( x = ( b ( a V( ; ab, = sup ( x ( x = ( b ( a (te Hece the ucto s o bouded varato o [ ab, ]. Smlarly a mootocally decreasg bouded ucto s o bouded varato wth total varato = ( a ( b. Thus or a bouded mootoc ucto V( = ( b ( a

2 Chap 7 Fuctos o bouded varato. Example A cotuous ucto may ot be a ucto o bouded varato. e.g. Cosder a ucto, where π xs ; whe < x ( x x ; whe x= It s clear that s cotuous o [,]. Let us choose the partto P =,,,...,,, The ( x ( x = ( ( π 3π 5π = sπ s + s s (+ π... + s + = = = Sce the te seres s dverget, thereore ts partal sums sequece { S }, where S = , s ot bouded above Thus ( x ( x ca be made arbtrarly large by takg sucetly large. V ;, ad so s ot o bouded varato. ( Remarks A ucto o bouded varato s ot ecessarly cotuous. e.g. the step-ucto ( x = [ x], where [ x ] deotes the greatest teger ot greater tha x, s a ucto o bouded varato o [,] but s ot cotuous. Theorem I the dervatve o the ucto exsts ad s bouded o [ ab, ], the s o bouded varato o [ ab, ]. s bouded o [ ab, ] k such that ( x k x [ ab, ]. Let P be ay partto o the terval [ ab, ] the ( x ( x = x x ( c, c [ ab, ] ( ; ab, kb a (by M.V.T V s te. s o bouded varato.

3 Chap 7 Fuctos o bouded varato. 3 Note Boudedess o s a sucet codto or V( to be te ad s ot ecessary. Theorem A ucto o bouded varato s ecessarly bouded. Suppose s o bouded varato o [ ab, ]. For ay x [ ab, ] axb,,, cosstg o just three pots the Aga, cosder the partto { } ( x ( a + ( b ( x V( ; ab, ( x ( a V( ; ab, ( x = ( a + ( x ( a ( a + ( x ( a ( ( a + V ; ab, < s bouded o [ ab, ]. Propertes o uctos o bouded varato The sum (derece o two uctos o bouded varato s also o bouded varato. Let ad g be two uctos o bouded varato o [ ab, ]. The or ay partto P o [ ab, ] we have ( + g ( x ( + g ( x = { ( x ( } { ( ( } + gx x + gx = ( x ( x + gx ( gx ( ( x ( x + gx ( gx ( V( ; ab, + V( gab ;, V( + gab ;, V( ; ab, + V( gab ;, Ths show that the ucto + g s o bouded varato. Smlarly t ca be show that g s also o bouded varato..e. V( g V( + V( g Note ( I ad g are mootoc creasg o [ ab, ] the ( g s o bouded varato o [ ab, ]. ( I c s costat, the sums ( x ( x ad thereore the total varato ucto, V( s same or ad c. The product o two uctos o bouded varato s also o bouded varato. Let ad g be two uctos o bouded varato o [ ab, ]. ad g are bouded ad a umber k such that ( x k & gx ( For ay partto P o [ ab, ] we have k x [ ab, ].

4 4 Chap 7 Fuctos o bouded varato. ( g ( x ( g ( x ( x gx ( ( x gx ( = = ( x gx ( ( x gx ( + ( x gx ( ( x gx ( { } { } = ( x gx ( gx ( + gx ( ( x ( x ( x gx ( gx ( + gx ( ( x ( x k gx ( gx ( + k ( x ( x kv( g + kv( g s o bouded varato o [ ab, ]. Note Theorems lke the above, could ot be appled to quotets o uctos because the recprocal o a ucto o bouded varato eed ot be o bouded varato. e.g. ( x as x x, the wll ot be bouded ad thereore ca ot ( x be o bouded varato o ay terval whch cotas x. Thereore to cosder quotet, we avod uctos whose values becomes arbtrarly close to zero. 3 I s a ucto o bouded varato o [ ab, ] ad a postve umber k such that ( x k x [ ab, ] the s also o bouded varato o [ ab, ]. For ay partto P o [ ab, ], we have ( x ( x = ( x ( x = ( x ( x ( x ( x k s o bouded varato o [ ab, ]. k ( x ( x V ( ; ab, 4 I s o bouded varato o [ ab, ], the t s also o bouded varato o [ ac, ] ad [, cb, ] where c s a pot o [ ab, ], ad coversely. Also V( ; ab, = V( ; ac, + V( ; cb,. a Let, rst, be o bouded varato o [ ab, ]. Take P = { a= x, x,..., xm = c} & P = { c= y, y,..., y = b} ay two parttos o [ ac, ] ad [, cb ] respectvely. Evdetly, P= P P = { a= x,..., xm, y,..., y = b} s a partto o [ ab, ]. We have m ( x ( x + ( y ( y V( ; ab,

5 Chap 7 Fuctos o bouded varato. 5 m ( ( x ( x V ; ab, ad ( y ( y V( ; ab, s o bouded varato o [ ac, ] ad [, cb ] both. b Let, ow, be o bouded varato o [ ac, ] ad [, cb ] both. Let P= { a= z, z,..., z = b} be a partto o [ ab, ]. * I t does ot cota the pot c, let us cosder the partto P = P {} c Let c [ z, r zr].e. z r c z, r < c r a z z The r z r ( z ( z r = ( z ( z + ( z ( z + ( z ( z r r r+ r ( z ( z + ( c ( z r + ( z ( c + ( z ( z r r+ ( ;, ( ;, ab t s o bouded varato o [, ] V ac + V cb s o bouded varato o [, ] [, cb ] both, the V( ; ab, V( ; ac, + V( ; cb,. ( Now let ε > be ay arbtrary umber. ;, ;, Sce V( ac ad V( cb are the total varato o o [, ] respectvely thereore partto P = { a= x, x, x,..., xm = c} ad P = { c= y y y y = b} o [ ac, ] & [, cb ] respectvely such that,, 3,..., m ε ( x ( x > V( ; ac,. ( ε & ( y ( y > V( ; cb, ( Addg ( ad ( we get m ac & ac & [, cb ] ( ( ( x ( x + ( y ( y > V ; ac, + V ; cb, ε ( ;, ( ;, ( ;, V ab > V ac + V cb ε But ε s arbtrary postve umber thereore we get V( ; ab, V( ; ac, + V( ; cb,.. (v From ( ad (v, we get V ; ab, = V ; ac, + V ; cb, ( ( (

6 6 Chap 7 Fuctos o bouded varato. Varato Fucto Let be a ucto o bouded varato o [ ab, ] ad x s a pot o [ ab, ]. The the total varato o s V( ; ax, o [ ax,, ] whch s clearly a ucto o x, s called the total varato ucto or smply the varato ucto o ad s deoted by V ( x, ad whe there s o scope or couso, t s smply wrtte as V( x. V ( x = V ; ax, ; ( a x b Thus ( I x, x are two pots o the terval [ ab, ] such that x > x, the ( x ( x V ; x, x V ( x V ( x ( ( ;, ( ;, = V ax V ax = V ( x V ( x mples that the varato ucto s mootocally creasg ucto o [ ab, ]. CHARACTERIZATION OF FUNCTIONS OF BOUNDED VARIATION Theorem A ucto o bouded varato s expressble as the derece o two mootocally creasg ucto. We have ( x = ( V( x + ( x ( V( x ( x = Gx ( H( x (say We shall prove that these two uctos Gx ( ad H( x are mootocally creasg o [ ab, ]. Now, x > x, we have Gx ( Gx ( = [ V( x V( x + ( x ( x ] = ( ;, ( ( ( V x x x x Sce V( ; x, x ( x ( x Gx ( Gx (.e. Gx ( Gx ( so that the ucto Gx ( s mootocally creasg o [ ab, ]. Aga, we have H( x H( x = ( V( x V( x ( ( x ( x = ( ;, ( ( ( V x x x x so that as beore H( x H( x.e. H( x H( x..e. H( x s also mootocally creasg ucto. Hece the result. Note A ucto ( x s o bouded varato over the terval [ ab, ] t ca be expressed as the derece o two mootocally uctos.

7 Chap 7 Fuctos o bouded varato. 7 Theorem Let be o bouded varato o [ ab, ]. Let V be deed o [ ab, ] as ollows: The ( V( x = V ( x = V ; ax, a< x b, V( a =. V s a creasg ucto o [ ab, ]. V s a creasg ucto o [ ab, ]. ( I a< x< y b, we ca wrte V( ; ay, = V( ; ax, V( ; xy, V( y V( x = V( ; xy, V( ; xy, a x V( y V( x V( x V( y ad ( holds. To prove (, let Dx ( = V( x ( x x [ ab, ]. The, a x< y b, we have Dy ( Dx ( = V( y V( x ( y ( x [ ] [ ] V( ; xy, [ ( y ( x ] = But rom the deto o V( ; xy,, t ollows that ( y ( x V( ; xy, Ths meas that D( y Dx ( ad ( holds. Theorem I c be ay pot o [ ab, ], the V( x s cotuous at c ad oly ( x s cotuous at c..e. A pot o cotuty o ( x s also a pot o cotuty o V( x ad coversely. Frstly suppose that V( x s cotuous at c. Let ε > be gve, the δ > such that V( x Vc ( < ε or x c < δ ( Also, we have ( x ( c V( x Vc ( x > c ( Ad ( x ( c Vc ( V( x x< c ( From (, ( ad (, we deduce that ( x ( c V( x Vc ( < ε or x c < δ Whch shows that ( x s cotuous at c. Now suppose that c s a pot o cotuty o ( x ad let ε > be gve, the δ > such that ( x ( c ε < or x c < δ P = c= y, y,..., y, y,..., y = b o [, cb ] such that Also a partto { q q } ( yq ( yq > V( ; cb, ε (v q= Sce as a result o troducg addto pots to the partto P, the correspodg sum o the modul o the dereces o the ucto values at ed pots wll ot be decreased, thereore we may assume that y b

8 8 Chap 7 Fuctos o bouded varato. < y c < δ ε so that ( y ( c <. (v Thus (v becomes V( ; cb, ε < ε + ( yq ( yq < ε + V( ; y, b q= V( ; cb, V( ; y, b < ε V( y V( c < ε Thus or < y c < δ, we have < V( y V( c < ε lm V( x = V( c x c+ Smlarly, we ca have lm V( x = Vc ( x c Whch shows that V( x s cotuous at c. Note V( x s cotuous [ ab, ] ( x s cotuous [ ab, ]. Corollary A ucto s o bouded varato o [ ab, ] there s a bouded creasg ucto g o [ ab, ] such that or ay two pots x ad x [ ab, ], x < x, we have ( x ( x gx ( gx ( Moreover, g s cotuous at x, so s. x V, Take ( a a< x b gx=, x= a The g s creasg ad bouded o [ ab, ]. Also, ( ( x x x Vx ( = gx ( gx ( Whch also yelds that g s cotuous at x, so s. Questo Show that the ucto deed by x s ( x = x,. s o bouded varato o [ ] ; x ; x= Soluto s deretable o [, ] ad ( x = x s s x or x. x Also ( x xs + sx + = 3 x.e. ( x, s bouded o [ ] Hece s o bouded varato o [, ].

9 Chap 7 Fuctos o bouded varato. 9 Questo πx xcos, < x Show that gx ( =, x= Soluto Let P =,,,...,,, 3 The ( x ( x s ot o bouded varato o [, ] be a partto o [, ]. = ( ( π π π π π π π = cos cos + cos cos + cos cos cos π π π π = cos + cos + cos cos π π π π = cos + cos + cos cos whch s ot bouded. Hece ( x s ot o bouded varato o [, ]. Alteratve We have gx ( gx ( + gx ( gx ( k+ k k k ( k + π kπ kπ ( k π = cos cos + cos cos k + k k k k = + k + k ; k s eve ; k s odd b Va ( g k= k k= k b s dverget Va ( g s ot te. k= k Hece g s ot o bouded varato. Reereces: ( Lectures & Notes (Year 3-4 Pro. Syyed Gul Shah Charma, Departmet o Mathematcs. Uversty o Sargodha, Sargodha. Made by: Atq ur Rehma (atq@mathcty.org Avalable ole at PDF Format. Page Setup: Legal ( 8 4 Prted: 3 Aprl 4 (Revsed: March 9, 6. Submt error or mstake at