Let $f: R \rightarrow R$ be the Signum Function defined as $f(x) = \begin{cases} 1, & x > 0 \\ 0, & x = 0 \\ -1, & x < 0 \end{cases}$ and $g: R \rightarrow R$ be the Greatest Integer Function given by $g(x) = [x]$,where $[x]$ is the greatest integer less than or equal to $x$. Do $fog$ and $gof$ coincide in $(0, 1]$?

(N/A) It is given that $f: R \rightarrow R$ is the Signum function and $g: R \rightarrow R$ is the Greatest Integer function $g(x) = [x]$.
For any $x \in (0, 1]$,we analyze the compositions $fog(x)$ and $gof(x)$.
First,consider $fog(x) = f(g(x)) = f([x])$.
If $x = 1$,then $g(1) = [1] = 1$,so $f(g(1)) = f(1) = 1$.
If $x \in (0, 1)$,then $g(x) = [x] = 0$,so $f(g(x)) = f(0) = 0$.
Thus,$fog(x) = \begin{cases} 1, & x = 1 \\ 0, & x \in (0, 1) \end{cases}$.
Next,consider $gof(x) = g(f(x))$.
Since $x \in (0, 1]$,$x > 0$,therefore $f(x) = 1$ for all $x \in (0, 1]$.
Thus,$gof(x) = g(1) = [1] = 1$ for all $x \in (0, 1]$.
Comparing the two,for $x \in (0, 1)$,$fog(x) = 0$ while $gof(x) = 1$.
Therefore,$fog$ and $gof$ do not coincide in $(0, 1]$.