In an infinite interval this is not true. But in a finite interval is this true? Or at least in a closed interval?
$\textbf{EDITED:} $ Ok, suppose that $ f:\left[ {a,b} \right] \to \mathbb{R} $ is Riemann integrable, is it true that the function $ \left| f \right|:\left[ {a,b} \right] \to \mathbb{R} $ is Riemann integrable? Where $ \left| f \right| $ denotes the function $ \left| {f\left( x \right)} \right| $ this is my first question, the other is with other kinds of finite length intervals, like open intervals, or semi-opens intervals.