Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12     1. Chapter 5 Class 12 Continuity and Differentiability
2. Serial order wise
3. Examples

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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 Function [x] Going by same Concept 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] Here, continuity will be measured at – 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 𝑥 =𝑑 if lim┬(x→𝑑) 𝑓(𝑥)= 𝑓(𝑑) Value of d can be 1.2, −3.2, 0.39 lim┬(x→𝑑) 𝑓(𝑥) = lim┬(x→𝑑) [𝑥] Putting x = d = [𝑑] 𝑓(𝑑) =[𝑑] Since lim┬(x→𝑑) 𝑓(𝑥)= 𝑓(𝑑) 𝑓(𝑥) 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 𝑥 =𝑐 if L.H.L = R.H.L = 𝑓(𝑐) if lim┬(x→𝑐^− ) 𝑓(𝑥)=lim┬(x→𝑐^+ ) " " 𝑓(𝑥)= 𝑓(𝑐) Value of c can be 1, −3, 0 LHL at x → c lim┬(x→𝑐^− ) f(x) = lim┬(h→0) f(c − h) = lim┬(h→0) [𝒄−𝒉] = lim┬(h→0) (𝒄−𝟏) = (𝑐−1) RHL at x → c lim┬(x→𝑐^+ ) g(x) = lim┬(h→0) g(c + h) = lim┬(h→0) [𝒄+𝒉] = lim┬(h→0) 𝐜 = 𝑐 Since LHL ≠ RHL ∴ f(x) is not continuous at x = c Hence, f(x) is not continuous at all integral points.

Examples 