Continuity and Differentiability · Question 2

JEE Advanced 2026 Paper 1NumericalMedium📈 Topic in 33% papers+4

Consider the function f : (- 2, 2 ) (- , ) defined by f(x) = (|x| + |x - 1|) x + [x x] , where [x x] is the greatest integer less than or equal to x x . Let be the total number of points in the interval (- 2, 2 ) at which f is NOT continuous, and let be the total number of points in the interval (- 2, 2 ) at which f is NOT differentiable. Then the value of + is .

Answer

Solution

x x is even, 0, increasing on [0, 2 ) from 0 to 2 1.57.

So [x x] jumps from 0 to 1 only where x x = 1, i.e. at x = x0 with x0 1.11; near x = 0 it stays 0.

The first part is continuous, so f is discontinuous only at x0: = 2.

Non-differentiable at x0 (discontinuous). |x| x is differentiable at 0 (behaves like x|x|), but |x - 1| x is not differentiable at x = 1 since 1 0 and 1 (- 2, 2 ).

= 3, so + = 5

Answer: 5

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