is integrable on a b f is bounded ie f x B for all x a b for some constant B

Is integrable on a b f is bounded ie f x b for all x

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Proof:Sincefis integrable on [a, b],fis bounded, i.e.,|f(x)| ≤Bfor allx[a, b] forsome constantB. Consider another functiong: [-B, B]R,t7→tn, which is continuouson [-B, B]. By Theorem above, the composed functiongf=fnmust be integrable on[a, b].Appendix. Some problems discussed in the classA. Iff(x)is a continuous function on[a, b]withf(x)0andf6=constant. ProveRbaf(x)dx >0.Proof:Sincef0 andf6=constant, there exists a pointc(a, b) such thatf(c)>0.By the continuity off, for any>0, there is aδ >0 such that|f(x)-f(c)|<,∀|x-c|< δ.By taking=f(c)2, we obtainf(x) =|f(x)|=|f(x)-f(c)+f(c)| ≥ |f(c)|-|f(x)-f(c)|> f(c)-f(c)2=f(c)2,∀|x-c|< δ.Then by Theorem 0.8, Corollary 0.6,Zbaf(x) =Zc-δaf(x)dx+Zc+δc-δf(x)dx+Zbc+δf(x)dx0 +Zc+δc-δf(x)dx+ 0f(c)22δ >0.7.2.2.True or false:(a) Iffis continuous on[0,2]and0f(x)4for allx[0,2], thenI=R20fexists and0I8.(b) Iffis integrable on[a, b]andgis integrable on[c, d], wheref([a, b])[c, d],thengfis integrable on[a, b].(c) Iffis integrable on[a, b], then|f|is integrable on[a, b]andRba|f| ≤ |Rbaf|.Solution:(a) True by the definition and by Corollay 0.6.(b) False. Letf(x) =(1q,ifx=pqQ[0,1], where(p, q) = 1;0,ifx[0,1]-Q.169
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Letg(x) =(0,ifx= 0;1ifx >0.Then by Theorem 0.10,gf(x) =(1,if xQ[0,1];0,ifx[0,1]-Q.Herefis integrable on [0,1],gis integrable on [0,+), butgfis not integrable on [0,1].
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  • Continuous function, Mj, F G, sup f

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