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Question Number 2289 by 123456 last updated on 14/Nov/15
suppose that f:[0,1]→R, lets f∈C^2   and suppose that ∃α∈[0,1] such that  f(α)+f(1−α)=1  proof or give a counter example that  ∃ξ∈[0,1],f(ξ)=ξ
$$\mathrm{suppose}\:\mathrm{that}\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R},\:\mathrm{lets}\:{f}\in\mathrm{C}^{\mathrm{2}} \\ $$$$\mathrm{and}\:\mathrm{suppose}\:\mathrm{that}\:\exists\alpha\in\left[\mathrm{0},\mathrm{1}\right]\:\mathrm{such}\:\mathrm{that} \\ $$$${f}\left(\alpha\right)+{f}\left(\mathrm{1}−\alpha\right)=\mathrm{1} \\ $$$$\mathrm{proof}\:\mathrm{or}\:\mathrm{give}\:\mathrm{a}\:\mathrm{counter}\:\mathrm{example}\:\mathrm{that} \\ $$$$\exists\xi\in\left[\mathrm{0},\mathrm{1}\right],{f}\left(\xi\right)=\xi \\ $$
Commented by Rasheed Soomro last updated on 15/Nov/15
THANK^S !  It′s increase in my knowledge.
$$\mathcal{THANK}^{\mathcal{S}} !\:\:{It}'{s}\:{increase}\:{in}\:{my}\:{knowledge}. \\ $$
Commented by RasheedAhmad last updated on 15/Nov/15
If  f:[0,1]→R means x∈[0,1] and f∈R  then what is the meaning of      f∈C^2  ?
$${If}\:\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R}\:{means}\:{x}\in\left[\mathrm{0},\mathrm{1}\right]\:{and}\:{f}\in\mathbb{R} \\ $$$${then}\:{what}\:{is}\:{the}\:{meaning}\:{of} \\ $$$$\:\:\:\:{f}\in\mathrm{C}^{\mathrm{2}} \:? \\ $$
Commented by prakash jain last updated on 15/Nov/15
f∈C^0   continuous function  f∈C^1   differentiable function and derivative               is continous  f∈C^2   double differentiable and second derivative               is continous.  similarly  f∈C^k  or f∈C^∞
$${f}\in\mathrm{C}^{\mathrm{0}} \:\:\mathrm{continuous}\:\mathrm{function} \\ $$$${f}\in\mathrm{C}^{\mathrm{1}} \:\:\mathrm{differentiable}\:\mathrm{function}\:\mathrm{and}\:\mathrm{derivative} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{is}\:\mathrm{continous} \\ $$$${f}\in\mathrm{C}^{\mathrm{2}} \:\:\mathrm{double}\:\mathrm{differentiable}\:\mathrm{and}\:\mathrm{second}\:\mathrm{derivative} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{is}\:\mathrm{continous}. \\ $$$$\mathrm{similarly} \\ $$$${f}\in\mathrm{C}^{{k}} \:\mathrm{or}\:{f}\in\mathrm{C}^{\infty} \\ $$

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