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Let-A-be-the-collection-of-functions-f-0-1-R-which-have-an-infinite-number-of-derivatives-Let-A-0-A-be-the-subcollection-of-those-functions-f-with-f-0-0-Define-D-A-0-A-by-D-f-df




Question Number 21463 by dioph last updated on 24/Sep/17
Let A be the collection of functions  f : [0, 1] → R which have an infinite  number of derivatives. Let A_0  ⊂ A  be the subcollection of those functions  f with f(0) = 0. Define D : A_0  → A  by D(f) = df/dx. Use the mean value  theorem to show that D is injective.  Use the fundamental theorem of  calculus to show that D is surjective.
LetAbethecollectionoffunctionsf:[0,1]Rwhichhaveaninfinitenumberofderivatives.LetA0Abethesubcollectionofthosefunctionsfwithf(0)=0.DefineD:A0AbyD(f)=df/dx.UsethemeanvaluetheoremtoshowthatDisinjective.UsethefundamentaltheoremofcalculustoshowthatDissurjective.

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