Let $[x]$ denote the greatest integer function. Then match List-I with List-II:

List-I	List-II
(A) $\vert x - 1\vert + \vert x - 2\vert$	(I) is differentiable everywhere except at $x = 0$
(B) $x - \vert x\vert$	(II) is continuous everywhere
(C) $x - [x]$	(III) is not differentiable at $x = 1$
(D) $x \, \vert x\vert$	(IV) is differentiable at $x = 1$

The function $|x - 1| + |x - 2|$ is the sum of two continuous absolute value functions. While it has corners at $x = 1$ and $x = 2$ where it is not differentiable, it remains continuous for all real values of $x$.

(A) matches with (II).

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$f(x) = x - |x|$ can be expressed as:

$f(x) = 0$ for $x \geq 0$

$f(x) = 2x$ for $x < 0$

The right-hand derivative at $x = 0$ is $0$, and the left-hand derivative is $2$. Because these values differ, the function is differentiable everywhere except at $x = 0$.

(B) matches with (I).

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The function $x - [x]$ represents the fractional part of $x$. The greatest integer function $[x]$ has jump discontinuities at every integer. At $x = 1$:

$\lim_{x \to 1^-} (x - [x]) = 1 - 0 = 1$

$\lim_{x \to 1^+} (x - [x]) = 1 - 1 = 0$

Since the function is discontinuous at $x = 1$, it is not differentiable at $x = 1$.

(C) matches with (III).

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The function $x|x|$ is defined as:

$f(x) = x^2$ for $x \geq 0$

$f(x) = -x^2$ for $x < 0$

For $x > 0$, the derivative is $2x$. At $x = 1$:

$f'(1) = 2(1) = 2$

The function is differentiable at $x = 1$.

(D) matches with (IV).

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The resulting matches are (A)-(II), (B)-(I), (C)-(III), (D)-(IV).