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

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Ex 5.1, 12 Find all points of discontinuity of f, where f is defined by 𝑓 𝑥﷯= 𝑥10−1, 𝑖𝑓 𝑥≤1﷮&𝑥2 , 𝑖𝑓 𝑥>1﷯﷯ We have 𝑓 𝑥﷯= 𝑥10−1, 𝑖𝑓 𝑥≤1﷮&𝑥2 , 𝑖𝑓 𝑥>1﷯﷯ Case 1 At x = 1 f is continuous at x = 1 if L.H.L = R.H.L = 𝑓 1﷯ i.e. if lim﷮x→ 1﷮−﷯﷯ 𝑓 𝑥﷯ = lim﷮x→ 1﷮+﷯﷯ 𝑓 𝑥﷯ = 𝑓 1﷯ Thus, L.H.L ≠ R.H.L ⇒ f is discontinuous at 𝒙 =𝟏 Case 2 Let x = c , where c < 1 ∴ 𝑓 𝑥﷯=𝑥10−1 f is continuous at x = c if lim﷮x→𝑐﷯ 𝑓 𝑥﷯=𝑓(𝑐) ⇒ f is continuous for 𝑥 =𝑐 less than 1. ⇒ f is at continuous for all real numbers less than 1. Case 3 Let x = c (where c > 1) 𝑓 𝑥﷯=𝑥2 f is continuous at x = c if lim﷮x→𝑐﷯ 𝑓 𝑥﷯=𝑓(𝑐) Thus lim﷮x→𝑐﷯ 𝑓 𝑥﷯=𝑓(𝑐) ⇒ f is continuous at 𝑥 =𝑐 (c is greater than 1) ⇒ f is continuous at all real numbers greater than 1. Hence, only x = 1 point of discontinuity ⇒ f is continuous for all real point except 1. Thus, f is continuous for all x∈R −{1}