UPSC NDA I 2022 Mathematics Solved Paper

Concept:The greatest integer function $y = \lfloor x \rfloor$ gives the largest integer $\le x$.It is constant on every open interval $(n, n+1)$ for integer $n$, so its derivative is $0$ there.It is discontinuous at every integer point, because left and right limits differ.Explanation:Statement 1: "Its derivative is $0$ at $x = 0.5$".Since $0.5$ lies in the open interval $(0,1)$, and $y = \lfloor 0.5 \rfloor = 0$ is constant there, the derivative exists and equals $0$.Thus statement 1 is correct.Statement 2: "It is continuous at $x = 0$".At $x = 0$, the left‑hand limit is $\lim\limits_{x \to 0^-} \lfloor x \rfloor = -1$ (since values between $-1$ and $0$ give $-1$).The right‑hand limit is $\lim\limits_{x \to 0^+} \lfloor x \rfloor = 0$ (values between $0$ and $1$ give $0$).The two limits are not equal, so the function is discontinuous at $x = 0$.Thus statement 2 is incorrect.Answer:Only statement 1 is correct. Option A: 1 only.