Function not piecewise continuous

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I would like to find a function $f$ such that $$ \left|\int_a^b f(x)\ dx \right| > \int_a^b |f(x)|\ dx $$ where the integral should be understand in the Riemann sense.

Of course, $f$ shouldn't be piecewise continuous.

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Your inequality is always false. For example see here for a proof of the opposite inequality.