Question

Suppose that f is integrable on (a, b) and define (f(x) if f(x) > 0 f+(x) = 3 and f (2)= if f(x) < 0, Show that f+ and f- are integrable on (a, b), and If(x) if f(x) > 0, if f(x) < 0. cb Sisleyde = [* p*(e) ds + [°r(a)di. | f(x) dx = | f+(x) dx + 1 f (x) dx.

Answer 1

Babelionas Given that 468C 'f' is integrable on {a,b] and Fid) - Flasif Fed) 20 if F(%) TO Flo) = { and if flosco Fias- so if fla120 For if flosko het 'P'card, ad.. . do b) be a partition of [a,b] and Z = [dos, dy], 8: 1,2, ..... Hexe we took, Mo: oup flas mr sup sco) m-inf flod, my = inf f (d) 11 when my to, me 20 then my my, my=mat and i Here, when M₂ 20 m2 30 then My = Myt, ez a my and Momz 2 M ing. and finally we have

i. When My 30 mg so, then we got My Myt: 0 m - mit and My - My 2 m d m > € 1mg - 3 Sg 2 & Miss E B Ź med og @ms, 5 % (1-3 Therepose the Equalon we gel is UCPF") -417.) SU(PP) -L(P2P). since ficou in integsable) olerable for a govern "630 These existe a partition 'p' such that UCP,F) -21P,F) <E foom the above Expocasion we have that U(P. 8+) - 2 (P, 8 ) <E Hence &rial is integoable on [a, b] we can also show that F-16) is integeable on [a, b] Sincc Flo= Ft (2) + F(d) and steze

integrable, F (a), F (d) are integrable on [a,b ] Here the sum of Integrable function is Now we have final Espsession ao fcom - f'(X) + fila) i integeable rence Socorda. S preos da + 5 Freas da li inteqoable on both (a,b).. Hence proved.