UPSC NDA I 2025 Mathematics Paper

Given,
The function is defined as:
f(x)={

x3,‌	‌ if ‌|x|<1
x2,‌	‌ if ‌|x|≥1

We are tasked with finding:

lim

x⟶−1

f′(x)
Check the left-hand limit for continuity at x=−1 :

lim

x⟶−1−

f(x)=(−1)2=1
Check the right-hand limit for continuity at x=−1 :

lim

x⟶−1+

f(x)=(−1)3=−1
Since the left-hand limit (L.H.S) and right-hand limit (R.H.S) are not equal, the function is discontinuous at x=−1.
Check the differentiability at x=1 :
The left-hand derivative at x=1 is L.H.D = 3 and the right-hand derivative at x=1 is R.H.D = 2 which means the function is not differentiable at x=1
∴ The function is neither continuous at x=−1 nor differentiable at x=1.