Limits, Continuity & Differentiability
Continuity and Differentiability
nta_pyq_2025_jan
Grade 12
Question:
Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function f (x) = [x] + |x - 2|, -2 < x < 3, is not continuous and not differentiable. Then m + n is equal to :
Step-by-Step Solution
Key Concept: Apply the core result for continuity and differentiability at a point and simplify using the given constraints.
(2) f (x) = [x] + |x - 2|, -2 < x < 3 ⎧ -x, -2 < x < -1 ⎪ ⎪ ⎪ ⎪ ⎪ 1 - x, -1 \le x < 0 ⎪ \therefore f (x) = ⎨ 2 - x, 0 \le x < 1 ⎪ ⎪ ⎪ 3 - x, 1 \le x < 2 ⎪ ⎪ ⎩ ⎪ x, 2 \le x < 3 It is clearly discontinues at 4 points and nondifferentiable at 4 points. \therefore m + n = 8
Correct Answer: 2