This is a qualifying exam problem from Indiana University.
Prove or provide a counterexample to the following statement:
If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, then there exists a real number $L$ such that
$\lim_{\epsilon \rightarrow 0} \int_{\epsilon \leq|x|\leq1} \frac{f(x)}{x}dx=L$
End of question.
I have shown that if f is differentiable at $0$ then is the statement is true, but I am having a hard time to find a counterexample with $f$ merely continuous or give a proof.
Is the statement true? Or can someone provide counterexample. Thanks.