f(x)={x|x|, amp;x0 0, amp; x=0. f(x)=0 f(0^ )=lim_n→0f(0 h)= 1 f(0^+)=1 So, f(x) is not continuous at x=0 f(x)={ 1, ...

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Bijective Function

fx=x|x|, x≠...

Question

$f (x) = ⎧ ⎨ ⎩ \begin{matrix} \frac{x}{| x |}, & x \neq 0 0, & x = 0 \end{matrix}$ .

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Solution

$f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} \frac{x}{| x |}, & x \neq 0 0, & x = 0 \end{matrix}$
$f (x) = 0$
$f (0^{-}) = {lim}_{n \to 0} f (0 - h) = - 1$
$f (0^{+}) = 1$
So, $f (x)$ is not continuous at $x = 0$
$f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} - 1, & x < 0 0, & x = 0 1, & x > 0 \end{matrix}$

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