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(C) We analyze each function in List $A$:$(A)$ $|x| + |x - 2|$: This is a sum of two continuous functions,so it is continuous for all real values of $

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$IV, II, V, I$

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(C) We analyze each function in List $A$:$(A)$ $|x| + |x - 2|$: This is a sum of two continuous functions,so it is continuous for all real values of $x$. This matches with $IV$.$(B)$ $	ext{cosech } x = \frac{2}{e^x - e^{-x}}$: This function is defined for all $x \in \mathbb{R} \setminus \{0\}$. Thus,it is continuous for all non-zero real values of $x$. This matches with $II$.$(C)$ $x - [x] = \{x\}$ (fractional part function): This function is known to be discontinuous at all integers. This matches with $V$.$(D)$ $\sqrt{2 - x}$: This function is defined for $x \le 2$. As $x 	o 2^+$,the expression under the square root becomes negative,so the right-hand limit does not exist in the real number system. This matches with $I$.Therefore,the correct matching is $A-IV, B-II, C-V, D-I$.

$A$. $\|x\| + \|x - 2\|$	$I$. Right hand limit does not exist at $x = 2$.
$B$. $\text{cosech } x$	$II$. Continuous only for non-zero real values of $x$.
$C$. $x - [x]$	$III$. Limit is zero for all real $x$.
$D$. $\sqrt{2 - x}$	$IV$. Continuous for all real value of $x$.
	$V$. Discontinuous at all integral values of $x$.

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