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JEE Advanced2026MathematicsContinuity and DifferentiabilityActual

For a real number , let [ ] denote the greatest integer less than or equal to . For a finite set S , let |S| denote the number of elements in the set S . Consider the functions f : (-3, 3) (- , ) and g : (-3, 3) (- , ) defined by f(x) = [x^3] _e(1 + ^2( (x - [x]))) and g(x) = x^3 ^2( _e(1 + x - [x])) . Let A = x (-3, 3) : f is discontinuous at x and B = x (-3, 3) : g is discontinuous at x . Then the value of |A| + 2|

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Step-by-step solution

For the function f(x) = [x^3] _e(1 + ^2( (x - [x]))) , we can simplify the argument of the sine function. Since [x] is an integer, ( (x - [x])) = ( x - [x]) = ( x) . Thus, ^2( (x - [x])) = ^2( x) . The function f(x) can be rewritten as f(x) = [x^3] _e(1 + ^2( x)) . The term _e(1 + ^2( x)) is continuous everywhere. The term [x^3] has jump discontinuities where x^3 is an integer. For x (-3, 3) , x^3 (-27, 27) . The integer values of x^3 in this interval are k -26, -25, , 26 , which gives 53 points of the form x = k^

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