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(D) Define the three functions: $g(x) = 1+x+[x]$,$h(x) = 2+x$,and $k(x) = x+2[x]$.For $x \in [0, 1)$: $g(x) = 1+x$,$h(x) = 2+x$,$k(x) = x$. Thus $f(x)

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$3$

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(D) Define the three functions: $g(x) = 1+x+[x]$,$h(x) = 2+x$,and $k(x) = x+2[x]$.For $x \in [0, 1)$: $g(x) = 1+x$,$h(x) = 2+x$,$k(x) = x$. Thus $f(x) = \max\{1+x, 2+x, x\} = 2+x$.For $x \in [1, 2)$: $g(x) = 2+x$,$h(x) = 2+x$,$k(x) = x+2$. Comparing $2+x$ and $x+2$,they are equal. Thus $f(x) = 2+x$.At $x = 2$: $g(2) = 1+2+2 = 5$,$h(2) = 2+2 = 4$,$k(2) = 2+2(2) = 6$. Thus $f(2) = \max\{5, 4, 6\} = 6$.So,$f(x) = 2+x$ for $x \in [0, 2)$ and $f(2) = 6$.Checking continuity: $\lim_{x 	o 2^-} f(x) = 2+2 = 4$,but $f(2) = 6$. Thus,$f$ is discontinuous at $x = 2$. So $m = 1$.Checking differentiability in $(0, 2)$: Since $f(x) = 2+x$ is a polynomial,it is differentiable for all $x \in (0, 2)$. So $n = 0$.Therefore,$(m+n)^2+2 = (1+0)^2+2 = 1+2 = 3$.

Let $[x]$ denote the greatest integer function and $f(x) = \max\{1+x+[x], 2+x, x+2[x]\}$ for $0 \leq x \leq 2$. Let $m$ be the number of points in $[0, 2]$ where $f$ is not continuous,and $n$ be the number of points in $(0, 2)$ where $f$ is not differentiable. Then $(m+n)^2+2$ is equal to:

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