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Given a continuous function $g:[a,b]\to\Bbb R$, if there exists a number $K>0$ s.t. for all $x\in[a,b]$, $|g(x)| \le K \int_a^x |g|$, prove $g(x)=0$ for all $x\in[a,b]$.

And I tried to derive some contradiction around $\inf g^{-1}(\Bbb R-\{0\})$ assuming $g\ne 0$, under given hypothesis, but I wasn't succesful.

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