Let f : R $\rightarrow$ R be the Signum Function defined as
$f(x) = \left\{ {\begin{array}{llllllllllllllllllll}{1,}&{x > 0}\\{0,}&{x = 0}\\{ - 1,}&{x < 0}\end{array}} \right.$ and g : R$\rightarrow$R be the Greatest Integer
Function given by g(x)$=$[x], where [x] is greatest integer less than or equal to x. Then, does fog and gof coincide in (0, 1] ?
For x $\in$ (0, 1]
(fog) (x) $=$ $f(g(x))$ $=$ $f([x])$
$= \left\{ {\begin{array}{llllllllllllllllllll}{f(0),}&{if}&{0 < x < 1}\\{f(1),}&{if}&{x = 1}\end{array}} \right. = \left\{ {\begin{array}{llllllllllllllllllll}{0,}&{if}&{0 < x < 1}\\{1,}&{if}&{x = 1}\end{array}} \right.$ …(1)
And (gof ) (x) $=$ g(f(x)) $=$g(1) $=$ [1] $=$ 1
$\Rightarrow$ $(gof)(x) = 1\forall x \in$ (0, 1] ...(2)
From (1) and (2), (fog ) and (gof ) do not coincide in (0, 1].
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