Checking continuity using LHL and RHL
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Example 15 Important You are here
Checking continuity using LHL and RHL
Last updated at April 16, 2024 by Teachoo
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±βπ ) π(π)= π(π ) L.H.S (π₯π’π¦)β¬(π±βπ ) π(π) = limβ¬(xβπ) [π₯] Putting x = d = [π] R.H.S 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.