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Given a function $f:\mathbb{R} \to \mathbb{R}$ such that it vanishes outside a compact interval $[-C,C]$. Its also known that $f$ is locally summable(absolutely integrable in every bounded region of $\mathbb{R}$).

I can neither prove it nor come up with a counterexample. Any hints on how to prove or disprove this?

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The function $f=(1/x^{1/2})\chi_{(0,1]}$ may work as a counterexample.

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