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Example 15 - Find all points of discontinuity of greatest integer - Examples

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Example 15 (Introduction) Find all the points of discontinuity of the greatest integer function defined by 𝑓 (𝑥) = [𝑥], where [𝑥] denotes the greatest integer less than or equal to 𝑥 Greatest Integer less than equal to 𝑥 Example 15 Find all the points of discontinuity of the greatest integer function defined by 𝑓 (𝑥) = [𝑥], where [𝑥] denotes the greatest integer less than or equal to 𝑥 Given f (x) = ﷐𝑥﷯ Let c be any real number f is continuous at x = c if LHL = RHL = f(c) i.e. ﷐﷐lim﷮𝑥→﷐𝑐﷮−﷯﷯﷮𝑓(𝑥)﷯ = ﷐﷐lim﷮𝑥→﷐𝑐﷮+﷯﷯﷮𝑓(𝑥)﷯ = f(c) Since ﷐﷐lim﷮𝑥→﷐𝑐﷮−﷯﷯﷮𝑓(𝑥)﷯ ≠ ﷐﷐lim﷮𝑥→﷐𝑐﷮+﷯﷯﷮𝑓(𝑥)﷯ ∴ f is discontinuous for all x ∈𝑹

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