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The number of points of discontinuity of the function f(x) = [2 x] - [ 2x] in the open interval x (0, ) , where [t] denotes the greatest integer less than or equal to t , is ________

Correct answer

5

Step-by-step solution

The function is f(x) = [2 x] - [ 2x] . Potential points of discontinuity in (0, ) occur where 2 x Z or 2x Z . For 2 x Z : Since x (0, ) , x (0, 1] , so 2 x (0, 2] . 2 x = 1 x = 1 2 x = 6 , 5 6 . 2 x = 2 x = 1 x = 2 . Let S₁ = 6 , 2 , 5 6 . For 2x Z : Since x (0, ) , 2x (0, 2 ) , so 2x [-1, 1) . 2x = 0 2x = 2 , 3 2 x = 4 , 3 4 . 2x = -1 2x = x = 2 . Let S₂ = 4 , 2 , 3 4 . The set of all potential points of discontinuity is S₁ S₂ = 6 , 4 , 2 , 3 4 , 5 6 . For all points except x = 2 , only one of the terms is an inte

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