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The number of points where the function f(x) = cases |x| + 1 & if x 0 [2 x] & if 0 is discontinuous, where [t] denotes the greatest integer t , is ______

Correct answer

4

Step-by-step solution

At x = 0 : _ x 0^- f(x) = _ x 0^- (|x| + 1) = 1 f(0) = 1 _ x 0^+ f(x) = _ x 0^+ [2 x] = 0 Since LHL RHL, f(x) is discontinuous at x = 0 . At x = : _ x ^- f(x) = _ x ^- [2 x] = 0 f( ) = | - | = 0 _ x ^+ f(x) = _ x ^+ |x - | = 0 Since LHL = RHL = f( ) , f(x) is continuous at x = . For x (0, ) , f(x) = [2 x] . The inner function g(x) = 2 x is continuous and its range on (0, ) is (0, 2] . Discontinuities of [g(x)] occur where g(x) crosses an integer. 2 x = 1 x = 6 , 5 6 . At these points, g(x) crosses 1 , so f(x) is di

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