Question

if F in R integrable on (a,b) then CF is R integrable on ___ where C=R???​

Answer 1

Answer:

1

Firstly let ϵ>0 be given. Then there exists a partition P of [a,b] such that U(p,f)−L(P,f)<ϵc

Let c>0 and  P be that partition of [a,b], and let mi and m′i be the infima of f and cf on some sub-interval Δxi of P.

Now, inf(cf)=c⋅inf(f)⟹c⋅ΔxiMi=M′iΔxi⟹c⋅L(P,f)=L(P,cf)

Similarly, let  Mi and  M′i be the suprema of f and cf, again on  Δxi. This now yields

Step-by-step explanation: