UPSC NDA I 2025 Mathematics Paper

Concept:Continuity requires the left and right limits to be equal at a point. Differentiability requires the left and right derivatives to be equal at a point.Explanation:The function is $f(x) = \begin{cases} x^3, & |x|<1 \\ x^2, & |x| \ge 1 \end{cases}$.For statement I at $x=-1$: Left-hand limit uses $x^2$ (since $|x|\ge1$), giving $(-1)^2=1$. Right-hand limit uses $x^3$ (since $|x|<1$ for $x>-1$), giving $(-1)^3=-1$. Limits differ, so the function is discontinuous at $x=-1$. Statement I is false.For statement II at $x=1$: Left-hand derivative from $x^3$ is $3x^2=3$. Right-hand derivative from $x^2$ is $2x=2$. Derivatives differ, so the function is not differentiable at $x=1$. Statement II is false.Thus neither statement is correct.Answer:D. Neither I nor II