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Let the funktions $f,g:\mathbb{R} \rightarrow \mathbb{R}$ and $a\in\mathbb{R}$.

a) $f$ is continuos in $a$ $\Leftrightarrow$ $|f|$ is continuos in a;

b) $f,g$ are continuos in $a$ $\Rightarrow$ $max\{f,g\}$ and $min\{f,g\}$ are continuos in $a$;

c) $f,g$ are continuos in $a$ $\Leftrightarrow$ $f \cdot g$ is continuos in $a$.

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    $\begingroup$ What are your thoughts? Hint: all claimed $\Leftarrow$ are false with simple counterexmples $\endgroup$ – Hagen von Eitzen Nov 20 '15 at 12:46
  • $\begingroup$ Please, thoroughly searched for an answer before asking your question here, and please read this $\endgroup$ – Nizar Nov 20 '15 at 13:54
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Hints:

(a) You may use the $\delta$-$\epsilon$ definition of continuity, the reverse triangle inequality.

(b) Here the first answer is a hint, and I think you can continue.

(c) Check this hint.

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