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Ex 5.1, 14 - Discuss continuity of f(x) = {3, 4, 5 if 0 < x < 1 - Checking continuity using LHL and RHL

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Ex 5.1, 14 Discuss the continuity of the function f, where f is defined by 𝑓 𝑥﷯= 3, 𝑖𝑓 0≤𝑥≤1﷮4, 𝑖𝑓 1<𝑥<3﷮ 5, 𝑖𝑓 3≤𝑥≤10﷯﷯ 𝑓 𝑥﷯= 3, 𝑖𝑓 0≤𝑥≤1﷮4, 𝑖𝑓 1<𝑥<3﷮ 5, 𝑖𝑓 3≤𝑥≤10﷯﷯ Case 1:- At x = 1 A function is continuous at x = 1 if L.H.L = R.H.L = 𝑓 1﷯ i.e. lim﷮x→ 1﷮−﷯﷯ 𝑓 𝑥﷯ = lim﷮x→ 1﷮+﷯﷯ = 𝑓 1﷯ Thus L.H.L ≠ R.H.L ⇒ lim﷮x→ 1﷮−﷯﷯ 𝑓 𝑥﷯ ≠ lim﷮x→ 1﷮+﷯﷯ = 𝑓 𝑥﷯ Hence 𝒇 𝒙﷯ is not continuous as 𝒙=𝟏 Case 2:- At x = 3 f is continuous at x = 3 if L.H.L = R.H.L = 𝑓 3﷯ i.e. lim﷮x→ 3﷮−﷯﷯ 𝑓 𝑥﷯ = lim﷮x→ 3﷮+﷯﷯𝑓 𝑥﷯ = 𝑓 3﷯ Since L.H.L ≠ R.H.L Hence 𝒇 𝒙﷯ is not continuous as 𝒙=𝟑 Case 3:- 0≤𝑥<1 𝑓 𝑥﷯ = 3 3 can be written as 𝑓 𝑥﷯ = 3. 𝑥﷮0﷯ So, f(x) is a polynomial We know that every polynomial function is continuous for every real number Therefore 𝒇 𝒙﷯ = 3 is continuous at 𝟎≤𝒙≤𝟏 . Case 4:- 1<𝑥<3 𝑓 𝑥﷯ = 4 4 can be written as 𝑓 𝑥﷯ = 4. 𝑥﷮0﷯ So, f(x) is a polynomial We know that every polynomial function is continuous for every real number Therefore 𝒇 𝒙﷯ = 3 is continuous at 𝟏<𝒙<𝟑 . Case 4:- 3 <𝑥≤ 10 𝑓 𝑥﷯ = 5 5 can be written as 𝑓 𝑥﷯ = 5. 𝑥﷮0﷯ So, f(x) is a polynomial We know that every polynomial function is continuous for every real number Therefore 𝒇 𝒙﷯ = 3 is continuous at 3 <𝒙≤ 10 Hence points of discontinuity are x = 1 & x = 3 Thus, f is continuous for all x ∈ R − 𝟏,𝟑﷯

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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