Question Number 1987 by 123456 last updated on 28/Oct/15
![lets a<b and f:[a,b]→R integable into [a,b] and continuous lets I a closed subset of [a,b] proof that (or give a conter example) ∫_I fdx=0 ∀I⊂[a,b]⇒f=0](https://www.tinkutara.com/question/Q1987.png)
$$\mathrm{lets}\:{a}<{b}\:\mathrm{and}\:{f}:\left[{a},{b}\right]\rightarrow\mathbb{R}\:\mathrm{integable}\:\mathrm{into}\:\left[{a},{b}\right]\:\mathrm{and}\:\mathrm{continuous} \\ $$$$\mathrm{lets}\:\mathrm{I}\:\mathrm{a}\:\mathrm{closed}\:\mathrm{subset}\:\mathrm{of}\:\left[{a},{b}\right] \\ $$$$\mathrm{proof}\:\mathrm{that}\:\left(\mathrm{or}\:\mathrm{give}\:\mathrm{a}\:\mathrm{conter}\:\mathrm{example}\right) \\ $$$$\underset{\mathrm{I}} {\int}{fdx}=\mathrm{0}\:\forall\mathrm{I}\subset\left[{a},{b}\right]\Rightarrow{f}=\mathrm{0} \\ $$
Commented by prakash jain last updated on 28/Oct/15
![If f is non−zero at some point c∈[a,b] then f is continuous⇒ ∃ ε>0 such that∫_(c−ε) ^(c+ε) fdx≠0](https://www.tinkutara.com/question/Q1992.png)
$$\mathrm{If}\:{f}\:\mathrm{is}\:\mathrm{non}−\mathrm{zero}\:\mathrm{at}\:\mathrm{some}\:\mathrm{point}\:{c}\in\left[{a},{b}\right]\:\mathrm{then} \\ $$$${f}\:\mathrm{is}\:\mathrm{continuous}\Rightarrow \\ $$$$\exists\:\epsilon>\mathrm{0}\:\mathrm{such}\:\mathrm{that}\underset{{c}−\epsilon} {\overset{{c}+\epsilon} {\int}}{fdx}\neq\mathrm{0} \\ $$