Someone can help me to find a counterexemple that the following proposition is not true:
$$\left | f \right |\in \mathcal{R}([a,b])\implies f\in \mathcal{R}([a,b])$$
Thank you so much for your help!
2026-04-04 04:39:32.1775277572
Counterexample for $\left | f \right |\in\mathcal{R}([a,b])\implies f\in \mathcal{R}([a,b])$
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Take $f(x)=\left\{\begin{matrix} 1 & x\in\mathbb{Q}\\ -1 & x\notin \mathbb{Q} \end{matrix}\right.$
Can you see why it works?