Ex 5.1, 19 - Show that g(x) = x - [x] is discontinuous at all - Ex 5.1

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Ex 5.1, 19 (Introduction) Show that the function defined by g (x) = x – [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x. Greatest Integer less than equal to 𝑥 Ex 5.1, 19 Show that the function defined by g (x) = x – [x] is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x. Given g(x) = x − 𝑥﷯ Let c be an integer. g (x) is continuous at x = c if L H L = RHL = g (c) i.e.. lim﷮𝑥→𝑐−﷯﷮𝑔 𝑥﷯= lim﷮𝑥→𝑐+﷯﷮𝑔 𝑥﷯=﷯﷯𝑔(𝑐) Hence, L H L ≠ RHL ∴ g (x) is not continuous at x = 6 Hence, g(x) is discontinuous at all integral points. Hence proved

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