Prove or find a counter example to those statements.

Question

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 $a$.

Answer 1

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.