Let $f : R \to R$ be the signum function defined as

$f (x) = ⎧ ⎨ ⎩ \begin{matrix} 1, & x > 0 0, & x = 0 - 1, & x < 0 \end{matrix}$

and $g : R \to R$ be the greatest
integer function given by $g (x) = [x]$ , where $[x]$ is greatest
integer less than or equal to $x$ . Then, does $f o g$ and $g o f$ coincide in
$(0, 1]$ ?

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Solution

$f (x) = ⎧ ⎨ ⎩ \begin{matrix} 1, & x > 0 0, & x = 0 - 1, & x < 0 \end{matrix} a n y = [x] f (x) = f [(x)] = ⎧ ⎨ ⎩ \begin{matrix} \begin{matrix} 0 & w h e n x \geq 0, x \leq 1 \end{matrix} \begin{matrix} 1 & w h e n x \geq 1 \end{matrix} \begin{matrix} - 1 & w h e n x < 0 \end{matrix} \end{matrix} f (x) = f (x) = ⎧ ⎨ ⎩ \begin{matrix} 1 x > 0 0 x = 0 - 1 x < 0 \end{matrix}$
in the interval $[- 1, 0]$
$f o g (x) = g o g (x) \forall \in [- 1, 0]$

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Let f:R→R be the signum function defined as f(x)=⎧⎨⎩1,x>00,x=0−1,x<0and g:R→R be the greatest integer function given by g(x)=[x], where [x] is greatestinteger less than or equal to x. Then, does fog and gof coincide in(0,1]?

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