Examples

Chapter 5 Class 12 Continuity and Differentiability
Serial order wise

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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 integerCase 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±βπ) π(π)= π(π) L.H.S (π₯π’π¦)β¬(π±βπ) π(π) = limβ¬(xβπ) [π₯] Putting x = d = [π] R.H.S π(π) =[π] Since limβ¬(xβπ) 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±βπ^β ) π(π)=(π₯π’π¦)β¬(π±βπ^+ ) " " π(π)= π(π) Value of c can be 1, β3, 0 LHL at x β c limβ¬(xβπ^β ) f(x) = limβ¬(hβ0) f(c β h) = limβ¬(hβ0) [πβπ] = limβ¬(hβ0) (πβπ) = (πβπ) RHL at x β c limβ¬(xβπ^+ ) g(x) = limβ¬(hβ0) g(c + h) = limβ¬(hβ0) [π+π] = limβ¬(hβ0) 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.