Let [x] denote the greatest integer less than or equal to x and x denote the fractional part of x . The number of points in the interval (-5, 5) where the function f(x) = ( x , [x]) is discontinuous, is _____.

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The function is given by f(x) = ( x , [x]) . We know that the fractional part x [0, 1) for all real x . For x 1 , [x] 1 > x , so f(x) = [x] . For x Thus, the function simplifies to: f(x) = x for x f(x) = [x] for x 1 The possible points of discontinuity are the integers in the interval (-5, 5) , which are -4, -3, -2, -1, 0, 1, 2, 3, 4 . For any integer k LHL: _ x k^- x = 1 RHL: _ x k^+ x = 0 So f(x) is discontinuous at x = -4, -3, -2, -1, 0 (5 points). For any integer k > 1 , f(x) = [x] in a neighborhood of k . LHL:

Let [x] denote the greatest integer less than or equal to x and x denote the fractional part of x . The number of points in the interval (-5, 5) where the function f(x) = ( x , [x]) is discontinuous, is _____.

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