Suppose that f is continuous on [a, b] and th | vedclass

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(D) Explanation: Let $f(x)$ be a continuous function on the interval $[a, b]$ such that $f(x) \in \mathbb{Z}$ for all $x \in [a, b]$.According to the

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$f(x)$ is constant

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(D) Explanation: Let $f(x)$ be a continuous function on the interval $[a, b]$ such that $f(x) \in \mathbb{Z}$ for all $x \in [a, b]$.According to the Intermediate Value Theorem,if $f$ is continuous on $[a, b]$,then $f$ must take all values between $f(a)$ and $f(b)$.Suppose $f$ is not constant. Then there exist $x_1, x_2 \in [a, b]$ such that $f(x_1) = n$ and $f(x_2) = m$ with $n 
eq m$. Without loss of generality,assume $n Since $f$ is continuous,for any value $y$ such that $n However,we are given that $f(x)$ is an integer for all $x \in [a, b]$.If we choose $y = n + 0.5$,which is not an integer,this contradicts the condition that $f(x)$ must be an integer.Therefore,the assumption that $f$ is not constant must be false.Thus,$f(x)$ must be a constant function.

Suppose that $f$ is continuous on $[a, b]$ and that $f(x)$ is an integer for each $x$ in $[a, b]$. Then in $[a, b]$

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