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Ex 5.1, 9 - Find all points of discontinuity - Chapter 5 Class 12 - Ex 5.1

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Ex 5.1, 9 Find all points of discontinuity of f, where f is defined by 𝑓﷐𝑥﷯=﷐﷐﷐𝑥﷮﷐𝑥﷯﷯, 𝑖𝑓 𝑥<0﷮&−1 , 𝑖𝑓 𝑥≥ 0﷯﷯ Given 𝑓﷐𝑥﷯=﷐﷐﷐𝑥﷮﷐𝑥﷯﷯, 𝑖𝑓 𝑥<0﷮&−1 , 𝑖𝑓 𝑥≥ 0﷯﷯ 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﷯ & 𝑓﷐0﷯ = − 1 Thus, L.H.L = R.H.L = f(0) ⇒ f is continuous at 𝑥 =0 Case 2 Let x = c (where c > 0) ∴ 𝑓﷐𝑥﷯=−1 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 for 𝑥 =(𝑐 greater than 0). ⇒ f is at continuous for all real numbers greater than 0. Case 3 Let x = c (where c < 0) 𝑓﷐𝑥﷯= ﷐𝑥﷮﷐𝑥﷯﷯ 𝑓﷐𝑥﷯=﷐𝑥﷮−𝑥﷯ 𝑓﷐𝑥﷯= − 1 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 for 𝑥 =𝑐 ( c is less than 0 ) ⇒ f is continuous for all real numbers less than 0. Thus, f is continuous for x ∈ R − {0} Hence f(x) is continuous at all points ⇒ f is continuous

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