Question

4a. (5 pts) Let f, g: [a, b -R be integrable. Show that la, blR, {f (x),g x)) h h (x) max and k[a, bR, k (x) min {f (x),g (x)) integrable. Hint: Observe that, for all a, b e R, max{a, b}= (a+ b+ la - bl) and min{a, b} (a+b-la -bl). are

Answer 1

Question set fog : 2963 er be integrable Show that hi [9,6] - ER h(x) = M04 | 1 (%) 9 » and k : [9,69 - R kls) = min { fenn geri? integrable. are Solusien given that fig are integrable and we write h(n) = Max & f(n) a g(x)} hin) = Źl f++ 1f-413 k(u) = ROM) = min & flag geno} Ź [+g-18-91) For proving hon) and . K(x) are integrable me prove Some theorems below as: 1.) let fi 59,6) OR 9: 19,6) v are integrable Then frg is integrable on [a,b] 2) set to [90] VR be integrable on [9,6) and ceR Then of is inreguable on [96] 3) ret a fun chion f: 59,61 o R is integrable on 19,6) Then It is integrable on [9.63 l

Now let us prove me above Stated theorem. 1.) let fag: 19,6] or be both integrable. on raibs Then ttg is integrable on 29,6). Proof let fogane integrable and bounded on [a,b] Then there exist positive real no. ka Rast I f(u) | IR and Igral Erg * ne [216]. walow I flu) +gr ml = | f(x)|+|gcm) le kitko ne [a,b] This fry are bounded on [9,6) let choose ESO Since f is said to be Riemann integrable on pai bj so there exis't portion P, on r9, 6] de UCPs f) - LCP, 9f) LE Same for g; UCP. 19) - LCR,J) Le FB partich - Bor 3 Konche. let Po = PUP be refinement of p, and Pz and LCP, f) = L(Poq f) < UC Pog f) « UCP, of) L(2,4) L(Pong) 4 U (Pong) < U(Ping) So Ul per f) - L(Pest) < Ulpin f) – LCP, f) << UC Pogg) - L(Pong) < U(B9) - L(BJ)< € set let Po= (Noong: ., Mn) where % 5% caca Mr = sup (f +g)(x) 5 mg - inf (fty) (x) ne [*0 %] me [my-,98) 4 . Sup flm) ne (no 1 any] et bred) m = int f(x) |

M" = sup gras me i nt gex) neln.9%) for r=1,2... Keres-9"] Then Mrs Ms + M", me m'tm," for 2 = 1, 2, N ( Poof+g) = M(, -80) + ... + Mn Can - 2n-1) (M' (^,-40) + ... + Mn' ( yn – *m-77 + M" (4, -40) +--+ Mn" Can-2n-1)] u Poa f+g) = u(fog f) + U(Pong) Similarly, L(Pos ft g) > L (Pos f) + LC Pogg) Hence U (Pon fxg) - L (Po if +g) < [u( Pos f) - L(Pos f)] le [U (Po, f) - L(Pong)) <ę + { =€ Therefore for any eso there exist a partition Po orf ra, b) suen that U(Pos fxg) - L (Po, fxg) <E This implies by sufficient condition for integrability ft g is integrable on [a, b] 2) Let fi 29,6] or be integrable on [9,6) and Cer then ct is integrable on [a, b] Proof since I is liemann integrable on [a,b] There cf is bounded on lab] Cose I C=0 С» т. с>0 Сое ш С <0

Care I Go Cf(x)=0 re (96) =) ct is integrable in [ab] Case II cro By definations St = supremum of set & LCPt): Pe partition ) of 19,63 } St . infimum of set { UCP, f): Pe Prably Now set the supremum of set of 2 (P, Cf): Pe Pran]} the Supremum of set Y CLCPF): PC P14,6]} - c. Supremum of sel P L Caitlipe P14,693 soi Since fin ieman integrable Riemann Puregrable on [a, b] se cf is Case Inczo same process for cose II So from -f Tues is theorem we have also reiemann integrable on Las)

3.) Let f: 19,67 OR be integrable on [96] Then It is integrable on (9,6] Proof Since f is reimann integrable on (9,6) so I fix) | ERV ne ra, b] kin any positive realnom If I is bounded on [96] Since f is integrable on (9,6] therefore there for any eso there exist pavention Papa = no, app en kn = 63 of carb) such rud ullf) - LCPP) ce till Now let my o My be the infimum of lfl on Dr = (no 19 %) and my' , My be true in finnum and supremum of If I on Fri For all a B e Ir' we have | 141 cm) - 11181] = 1 14(a) 1 - 1 f(e)l). < 1 fm) – f($) < My - my My' - mýs My - My ra 1,2.... so UCPs | fl- L(B141) = E CM,_m) Sr = 3 (Mm), = UP, f ) - LC8+) <E. so Iflis Riemann integrable on [ 9,6).

Thos using Theorem ☺ ☺☺ meget of f and I are inregeable on [a, b] then ttg is also integrable on 59, 6] and fuga I f-gl is also integrable on 19,63 so so (ftg) + 1 f-gl and (f +g) - 1 fugl is integrable on [916) we proved hens = [cftg) + 1f-31] k(n) = t[ (ftg) - 1f-gl] integrable on [a,b], is