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find-or-prove-it-can-t-exist-a-f-R-R-diferentiable-such-that-a-a-f-x-dx-0-a-R-gt-0-df-dx-0-x-R-




Question Number 56205 by 121194 last updated on 12/Mar/19
find (or prove it can′t exist) a f:R→R diferentiable  such that  ∫_(a−δ) ^(a+δ) f(x)dx=0,∀a∈R,δ>0  (df/dx)=0,∀x∈R
$$\mathrm{find}\:\left(\mathrm{or}\:\mathrm{prove}\:\mathrm{it}\:\mathrm{can}'\mathrm{t}\:\mathrm{exist}\right)\:\mathrm{a}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{diferentiable} \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$$\underset{{a}−\delta} {\overset{{a}+\delta} {\int}}{f}\left({x}\right){dx}=\mathrm{0},\forall{a}\in\mathbb{R},\delta>\mathrm{0} \\ $$$$\frac{{df}}{{dx}}=\mathrm{0},\forall{x}\in\mathbb{R} \\ $$
Answered by MJS last updated on 12/Mar/19
f(x)=0
$${f}\left({x}\right)=\mathrm{0} \\ $$

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