Ex 5.1, 15 - Discuss continuity of f(x) = {2x, 0, 4x - Class 12 - Ex 5.1

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  1. Chapter 5 Class 12 Continuity and Differentiability
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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. lim﷮x→ 0﷮−﷯﷯ 𝑓 𝑥﷯ = lim﷮x→ 0﷮+﷯﷯ = 𝑓 𝑥﷯ = 𝑓 0﷯ i.e. lim﷮x→ 0﷮−﷯﷯ 𝑓 𝑥﷯ = lim﷮x→ 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. lim﷮x→ 1﷮−﷯﷯ 𝑓 𝑥﷯ = lim﷮x→ 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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