Ex 5.1 ,5 Is f(x) = {x x <=1, 5 x > 1 continuous at x = 0, 1, 2 - Checking continuity at a given point

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Ex 5.1, 5 Is the function f defined by 𝑓﷐𝑥﷯=﷐﷐𝑥, 𝑖𝑓 𝑥≤1﷮&5, 𝑖𝑓 𝑥>1﷯﷯ continuous at 𝑥 = 0 ? At 𝑥 = 1 ? At 𝑥 = 2 ? Given 𝑓﷐𝑥﷯=﷐﷐𝑥, 𝑖𝑓 𝑥≤1﷮&5, 𝑖𝑓 𝑥>1﷯﷯ At 𝑥 = 0 f is continuous at x = 0 if ﷐lim﷮x→0﷯ 𝑓﷐𝑥﷯= 𝑓﷐0﷯ Hence ﷐lim﷮x→0﷯ 𝑓﷐𝑥﷯=𝑓(0) ⇒ f is continuous at x=0. At 𝒙 = 𝟏 𝑓﷐𝑥﷯=﷐﷐𝑥, 𝑖𝑓 𝑥≤1﷮&5, 𝑖𝑓 𝑥>1﷯﷯ f 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﷯=1 Since L.H.L ≠ R.H.L ⇒ f is discontinuous at x=1. At 𝒙 = 𝟐 f is continuous at x = 2 If ﷐lim﷮x→2﷯ 𝑓﷐𝑥﷯= 𝑓﷐2﷯ Hence, ﷐lim﷮x→𝑐﷯ 𝑓﷐𝑥﷯=𝑓﷐2﷯ Hence f is continuous at x=2.

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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.