Last updated at Aug. 19, 2021 by Teachoo

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Example 15 (Introduction) Find all the points of discontinuity of the greatest integer function defined by π (π₯) = [π₯], where [π₯] denotes the greatest integer less than or equal to π₯Greatest Integer Function [x] Going by same Concept Example 15 Find all the points of discontinuity of the greatest integer function defined by π(π₯) = [π₯], where [π₯] denotes the greatest integer less than or equal to π₯Given π(π₯) = [π₯] Here, Continuity will be measured at β integer numbers, and non integer numbers Thus, we check continuity for When x is an integer When x is not an integer Case 1 : When x is not an integer f(x) = [x] Let d be any non integer point Now, f(x) is continuous at π₯ =π if (π₯π’π¦)β¬(π±βπ ) π(π)= π(π ) Value of d can be 1.2, β3.2, 0.39 (π₯π’π¦)β¬(π±βπ ) π(π) = limβ¬(xβπ) [π₯] Putting x = d = [π] π(π ) =[π] Since limβ¬(xβπ) π(π₯)= π(π) π(π₯) is continuous for all non-integer points Case 2 : When x is an integer f(x) = [x] Let c be any non integer point Now, f(x) is continuous at π₯ =π if L.H.L = R.H.L = π(π) if (π₯π’π¦)β¬(π±βπ^β ) π(π)=(π₯π’π¦)β¬(π±βπ^+ ) " " π(π)= π(π) Value of c can be 1, β3, 0 LHL at x β c limβ¬(xβπ^β ) f(x) = limβ¬(hβ0) f(c β h) = limβ¬(hβ0) [πβπ] = limβ¬(hβ0) (πβπ) = (πβπ) RHL at x β c limβ¬(xβπ^+ ) g(x) = limβ¬(hβ0) g(c + h) = limβ¬(hβ0) [π+π] = limβ¬(hβ0) π = π Since LHL β RHL β΄ f(x) is not continuous at x = c Thus, we can say that f(x) is not continuous at all integral points.

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Chapter 5 Class 12 Continuity and Differentiability (Term 1)

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.