Ex 5.1 ,6 - Find all points of discontinuity of f(x) = {2x + 3 - Ex 5.1

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Ex 5.1, 6 Find all points of discontinuity of f, where f is defined by 𝑓﷐𝑥﷯=﷐﷐2𝑥+3, 𝑖𝑓 𝑥≤2﷮&2𝑥−3, 𝑖𝑓 𝑥>2﷯﷯ We have, 𝑓﷐𝑥﷯=﷐﷐2𝑥+3, 𝑖𝑓 𝑥≤2﷮&2𝑥−3, 𝑖𝑓 𝑥>2﷯﷯ Case 1 At 𝑥 =2 f is continuous at x = 2 if L.H.L = R.H.L = 𝑓﷐2﷯ i.e. ﷐lim﷮x→﷐2﷮−﷯﷯ 𝑓﷐𝑥﷯=﷐lim﷮x→﷐2﷮+﷯﷯ 𝑓﷐𝑥﷯=𝑓﷐2﷯ Since, L.H.L ≠ R.H.L ∴ f is not continuous at x=2. Case 2 At 𝑥 =𝑐 where c < 2 𝑓﷐𝑥﷯= 2𝑥+3 f is continuous at x=c if ﷐lim﷮x→𝑐﷯ 𝑓﷐𝑥﷯=𝑓﷐𝑐﷯ Hence, ﷐lim﷮x→𝑐﷯ 𝑓﷐𝑥﷯=𝑓﷐𝑐﷯ ∴ f is continuous at x=c where c<2 Thus, f is continuous at all real number less than 2. Case 3 At 𝑥 =𝑐 where c > 2 ∴ 𝑓﷐𝑥﷯= 2 𝑥+3 f is continuous at x=c if ﷐lim﷮x→𝑐﷯ 𝑓﷐𝑥﷯=𝑓﷐𝑐﷯ Hence, ﷐lim﷮x→𝑐﷯ 𝑓﷐𝑥﷯=𝑓﷐𝑐﷯ ⇒ f is continuous at x=c where c>2 ⇒ f is continuous at all real number greater than 2 Hence, only x=2 is point is discontinuity. ⇒ f is continuous at all real numbers except 2. Thus, f is continuous for 𝐱 ∈ R − {2}.

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