# Ex 5.1, 15 - Chapter 5 Class 12 Continuity and Differentiability

Last updated at Sept. 17, 2019 by Teachoo

Last updated at Sept. 17, 2019 by Teachoo

Transcript

Ex 5.1, 15 Discuss the continuity of the function f, where f is defined by 𝑓 𝑥= 2𝑥, 𝑖𝑓 𝑥<0 0, 𝑖𝑓 0≤𝑥≤1 4𝑥, 𝑖𝑓 𝑥>1 Case 1:- At x = 0 𝑓 is continuous at x = 0 if L.H.L = R.H.L = 𝑓 0 i.e. limx→ 0− 𝑓 𝑥 = limx→ 0+ = 𝑓 𝑥 = 𝑓 0 i.e. limx→ 0− 𝑓 𝑥 = limx→ 0+ = 𝑓 𝑥 = 𝑓 0 & 𝑓(𝑥) = 0 𝑓(0) = 0 Thus L.H.L = R.H.L Hence 𝒇 𝒙 is not continuous at 𝒙=𝟎 Case 2: At x = 1 𝑓 is continuous at x = 1 if if L.H.L = R.H.L = 𝑓 1 i.e. limx→ 1− 𝑓 𝑥 = limx→ 1+ = 𝑓 𝑥 = 𝑓 1 Thus L.H.L ≠ R.H.L Hence, 𝑓 𝑥 is not continuous at 𝒙=𝟏 Case 3:- 0≤𝑥<1 𝑓 𝑥 = 0 Since 𝑓 𝑥 is a constant function, it is continuous. ∴ 𝑓 𝑥 is continuous at 𝟎≤𝒙<𝟏 Case 4:- For 𝑥<0 𝑓 𝑥 = 2𝑥 𝑓 𝑥 is continuous, as it is a polynomial. Hence 𝒇 𝒙 is continuous for all real number less then 0 Case 5:- 𝑥>1 𝑓 𝑥 = 4𝑥 𝑓 𝑥 is continuous, as it is a polynomial. Hence 𝒇 𝒙 is continuous for all real number greater then 1 Hence points of discontinuity are x = 1 Thus, f is continuous for all x ∈ R − 𝟏

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.