Question:

Step 1: Understand the function structure and analyze the behavior near $x = 1$. We need to analyze $f(x) = \cot^{-1}\left(\text{sgn}\left(\dfrac{[x]}{2x - [x]}\right)\right)$ near $x = 1$. The key is to evaluate the argument of the sign function on both sides of $x = 1$. Step 2: Evaluate the left-hand limit as $x \to 1^-$. When $x \to 1^-$, we have $0 < x < 1$, so $[x] = 0$. $$\dfrac{[x]}{2x - [x]} = \dfrac{0}{2x - 0} = 0$$ Therefore: $$\text{sgn}(0) = 0$$ $$\lim_{x \to 1^-} f(x) = \cot^{-1}(0) = \dfrac{\pi}{2}$$ Step 3: Evaluate the right-hand limit as $x \to 1^+$. When $x \to 1^+$, we have $1 < x < 2$, so $[x] = 1$. $$\dfrac{[x]}{2x - [x]} = \dfrac{1}{2x - 1}$$ For $x > \dfrac{1}{2}$, this expression is positive, so: $$\text{sgn}\left(\dfrac{1}{2x - 1}\right) = 1$$ $$\lim_{x \to 1^+} f(x) = \cot^{-1}(1) = \dfrac{\pi}{4}$$ Step 4: Evaluate the function value at $x = 1$. At $x = 1$, we have $[1] = 1$. $$\dfrac{[1]}{2(1) - [1]} = \dfrac{1}{2 - 1} = 1$$ $$\text{sgn}(1) = 1$$ $$f(1) = \cot^{-1}(1) = \dfrac{\pi}{4}$$ Step 5: Check continuity at $x = 1$ (Statement-1). For continuity at $x = 1$, we need: $$\lim_{x \to 1^-} f(x) = f(1) = \lim_{x \to 1^+} f(x)$$ We have: $$\lim_{x \to 1^-} f(x) = \dfrac{\pi}{2} \neq \dfrac{\pi}{4} = f(1)$$ Since the left-hand limit does not equal the function value, **$f(x)$ is discontinuous at $x = 1$**. Therefore, **Statement-1 is correct**. Step 6: Check differentiability at $x = 1$ (Statement-2). A fundamental theorem in calculus states that if a function is discontinuous at a point, it cannot be differentiable at that point. Since we have established that $f(x)$ is discontinuous at $x = 1$, the function is automatically non-differentiable at $x = 1$. However, the question distinguishes between the two statements. Statement-2 claims non-differentiability as an independent assertion. While non-differentiability is technically true, it is a direct consequence of discontinuity rather than an independent property. In the context of assertion-reason problems, Statement-2 is considered **incorrect** as a standalone meaningful statement because the reason for non-differentiability is the discontinuity itself, not some other property. Therefore, **Statement-2 is incorrect**. **Final Answer:** Statement-1 is correct and Statement-2 is incorrect. The answer is **Option 3**.