Ex 5.1, 16 - Discuss continuity of f(x) = { -2, 2x, 2 - Chapter 5 - Ex 5.1

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Ex 5.1, 16 Discuss the continuity of the function f, where f is defined by 𝑓 𝑥﷯= −&2, 𝑖𝑓 𝑥≤−1﷮2&𝑥, 𝑖𝑓 −1≤𝑥≤1﷮2, 𝑖𝑓 𝑥>1 ﷯﷯ 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 𝑥﷯ = 𝑓 −1﷯ And 𝑓 −2﷯ = −2 Thus L.H.L = R.H.L = 𝑓 −2﷯ = −2 Hence 𝒇 𝒙﷯ is continuous at x = −𝟏 Case 1:- At x = −1 A function is continuous at x = − 1 if A function is continuous at x = 1 if if L.H.L = R.H.L = 𝑓 1﷯ i.e. lim﷮x→ 1﷮−﷯﷯ 𝑓 𝑥﷯ = lim﷮x→ 1﷮+﷯﷯ 𝑓 𝑥﷯ = 𝑓 1﷯ & 𝑓 1﷯ = 2 1﷯ = 2 Thus L.H.L = R.H.L = 𝑓 −2﷯ Hence 𝒇 𝒙﷯ is continuous at x = 1 Case 3:- For 𝑥<−1 𝑓 𝑥﷯ = −2 Thus, 𝑓 𝑥﷯ is a constant function . & Every constant function is continuous for all real number. Hence 𝒇 𝒙﷯ is continuous at 𝒙<−𝟏 Case 4:- For 𝑥>1 𝑓 𝑥﷯ = 2 Thus, 𝑓 𝑥﷯ is a constant function . & Every constant function is continuous for all real number. Hence 𝒇 𝒙﷯ is continuous at 𝒙>𝟏 Case 5:- For −1≤𝑥≤1 𝑓 𝑥﷯ = 2𝑥 So, f(x) is a polynomial & Every polynomial is continuous. ⇒ 𝒇 𝒙﷯ is continuous at −𝟏<𝒙≤𝟏 Thus, f(x) is continuous for all real numbers, i.e. f is continuous for all x ∈ R

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.