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JEE Advanced2026MathematicsContinuity and DifferentiabilityActual

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 ___________.

Correct answer

0

Step-by-step solution

Let f(x) = g(x) + h(x) , where g(x) = (|x| + |x - 1|) x and h(x) = [x x] . First, we analyze h(x) = [x x] on the interval (- 2 , 2 ) . The function y = x x is an even function. For x [0, 2 ) , x x is strictly increasing. At x = 0 , x x = 0 . As x 2 , x x 2 1.57 . Thus, x x [0, 1.57) for x (- 2 , 2 ) . The value of [x x] will be 0 when 0 x x Let x₁ (0, 2 ) be the unique point where x₁ x₁ = 1 . By symmetry, -x₁ is the unique point in (- 2 , 0 ) where (-x₁) (-x₁) = 1 . Therefore, h(x) has jump discontinuities at exact

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