Continuity and Differentiability · Question 14

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The number of points of discontinuity of the function f(x) = [ x22 ] - [ x], x [0, 4], where [ ] denotes the greatest integer function, is .

Answer

Solution

[ x22 ] jumps where x22 = 1, 2, , 8, i.e. x = 2, 2, 6, 2 2, 10, 2 3, 14, 4

[ x] jumps where x = 1, 2, i.e. x = 1, 4

At x = 4 (right end point, only the left-hand limit matters):

x 4- f(x) = 7 - 1 = 6 and f(4) = 8 - 2 = 6 f is continuous at x = 4.

At x = 0: f(x) = 0 for 0 x < 1 continuous.

At each of x = 1, 2, 2, 6, 2 2, 10, 2 3, 14 only one of the two terms jumps f is discontinuous there.

number of points of discontinuity = 8

Answer: 8

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