Match List-I with List-II. Here [x] denotes the greatest integer function

Choose the correct answer from the options given below:

(A)-(II), (B)-(III), (C)-(IV), (D)-(I)

The correct answer is Option (2) → (A)-(II), (B)-(III), (C)-(IV), (D)-(I)

$\textbf{(A)}\; f(x)=\lfloor x \rfloor$

The greatest–integer function has jump discontinuities at every integer.

So it matches: “continuous everywhere except at all integral values.”

$A \to (II)$

$\textbf{(B)}\; f(x)=|x-1|$

Absolute value functions are continuous everywhere but not differentiable where the inside becomes zero → at $x=1$.

So it matches: “continuous everywhere but not differentiable at $x=1$.”

$B \to (III)$

$\textbf{(C)}\; f(x)=e^{|x|}$

Continuous everywhere, but $|x|$ is not differentiable at $x=0$, so $e^{|x|}$ is also not differentiable at $x=0$.

So it matches: “continuous everywhere but not differentiable at $x=0$.”

$C \to (IV)$

$\textbf{(D)}\; f(x)=|x+1|$

Non-differentiable where $x+1=0$ → at $x=-1$.

So it matches: “continuous everywhere but not differentiable at $x=-1$.”

$D \to (I)$

$\textbf{Final Answer: } A\to II,\; B\to III,\; C\to IV,\; D\to I$