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Question Number 21463 by dioph last updated on 24/Sep/17

$$\mathrm{Let}A\mathrm{be}\mathrm{the}\mathrm{collection}\mathrm{of}\mathrm{functions}$$$$f:[0,1]\to \mathbb{R}\mathrm{which}\mathrm{have}\mathrm{an}\mathrm{infinite}$$$$\mathrm{number}\mathrm{of}\mathrm{derivatives}.\mathrm{Let}{A}_0\subset A$$$$\mathrm{be}\mathrm{the}\mathrm{subcollection}\mathrm{of}\mathrm{those}\mathrm{functions}$$$$f\mathrm{with}f\left(0\right)=0.\mathrm{Define}D:A_0\to A$$$$\mathrm{by}D\left(f\right)=df/dx.\mathrm{Use}\mathrm{the}\mathrm{mean}\mathrm{value}$$$$\mathrm{theorem}\mathrm{to}\mathrm{show}\mathrm{that}D\mathrm{is}\mathrm{injective}.$$$$\mathrm{Use}\mathrm{the}\mathrm{fundamental}\mathrm{theorem}\mathrm{of}$$$$\mathrm{calculus}\mathrm{to}\mathrm{show}\mathrm{that}D\mathrm{is}\mathrm{surjective}.$$

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