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Let $f(x)$ be a Lebesgue integrable function. Then is it true that $$ \lim_{\epsilon\to 0}\int_0^\epsilon f(x)\,dx=0 $$ always? When $f(x)$ is bounded answer is trivial, but if we wish to show this for unbounded functions, how would one proceed?

Andrey Carl
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If $f$ is integrable, then this follows from either the monotone or dominated convergence theorems, with dominating function $|f|$.