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Question Number 1987 by 123456 last updated on 28/Oct/15

$$\mathrm{lets}a<b\mathrm{and}f:[a,b]\to \mathbb{R}\mathrm{integable}\mathrm{into}[a,b]\mathrm{and}\mathrm{continuous}$$$$\mathrm{lets}\mathrm{I}\mathrm{a}\mathrm{closed}\mathrm{subset}\mathrm{of}[a,b]$$$$\mathrm{proof}\mathrm{that}\left(\mathrm{or}\mathrm{give}\mathrm{a}\mathrm{conter}\mathrm{example}\right)$$$$\underset{\mathrm{I}}{\int }fdx=0\mathrm{\forall }\mathrm{I}\subset [a,b]\Rightarrow f=0$$

Commented by prakash jain last updated on 28/Oct/15

$$\mathrm{If}f\mathrm{is}\mathrm{non}-\mathrm{zero}\mathrm{at}\mathrm{some}\mathrm{point}c\in [a,b]\mathrm{then}$$$$f\mathrm{is}\mathrm{continuous}\Rightarrow$$$$\mathrm{\exists }ϵ>0\mathrm{such}\mathrm{that}\underset{c-ϵ}{\stackrel{c+ϵ}{\int }}fdx\ne 0$$

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