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Is it true that if f continuous and goes to $0$ at $\infty$ then f is Lebesgue function?

I don't know how to prove it.

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No. Consider $f(x)=\cases{1,&$ -1$\cr 1/|x|, &\text{otherwise} }$.

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    But probably, someone more clever than myself can come up with a simpler counterexample...2012-12-08