Question

(5) Let qe Q. Suppose that a <b, 0<c<d, and that f : [a, b] → [c, d]. If f is integrable on [a,b], then prove that * (t)dt) = f'(x) for all 3 € (a, b).

Answer 1

Solution '- Pooue that SY the Pooper tiel of integoals we Knaw that P is Riemamm imbegoable on mLa,b]=D>1f|i8 Riemam integgable [a,b}. P i8 bounded on {a, b] implies Ie is bamded on fa,b} => Le1 - p ;8 bounded on fa, 6} since f"- IfI" Since q EQ Oud it i8 toul for f 20 00fco Since -f also bounded on {a,63. without loss of generality we can suPPose that f>0 Iet supf in Sa,b] Let &>0, feRfa,b? implies there sxist8 PE f la,b3 Such that E (My -m)&, -0 (P,e)-L(P,f)< 2M+1 im I,, imf (p) = m and Sup (f)- m uLP,p")-L(P,P")- (m-)S, -E (My-ms) (m -m V-2

(Mo-m,) (MaM)8 c 2M 2M+1 . FOO Soch &>0 Coe con find PE fla,bs Such that ulP, p^)_L(P,pa) <s :p ;8 integrable on fa,b7 BY the de Pinition of fiost mean value theasem f°(x). NOW S P(t) dt VI