1. Chapter 5 Class 12 Continuity and Differentiability
2. Serial order wise
3. Ex 5.1

Transcript

Ex 5.1, 12 Find all points of discontinuity of f, where f is defined by ๐ ๐ฅ๏ทฏ= ๐ฅ10โ1, ๐๐ ๐ฅโค1๏ทฎ&๐ฅ2 , ๐๐ ๐ฅ>1๏ทฏ๏ทฏ We have ๐ ๐ฅ๏ทฏ= ๐ฅ10โ1, ๐๐ ๐ฅโค1๏ทฎ&๐ฅ2 , ๐๐ ๐ฅ>1๏ทฏ๏ทฏ Case 1 At x = 1 f is continuous at x = 1 if L.H.L = R.H.L = ๐ 1๏ทฏ i.e. if lim๏ทฎxโ 1๏ทฎโ๏ทฏ๏ทฏ ๐ ๐ฅ๏ทฏ = lim๏ทฎxโ 1๏ทฎ+๏ทฏ๏ทฏ ๐ ๐ฅ๏ทฏ = ๐ 1๏ทฏ Thus, L.H.L โ  R.H.L โ f is discontinuous at ๐ =๐ Case 2 Let x = c , where c < 1 โด ๐ ๐ฅ๏ทฏ=๐ฅ10โ1 f is continuous at x = c if lim๏ทฎxโ๐๏ทฏ ๐ ๐ฅ๏ทฏ=๐(๐) โ f is continuous for ๐ฅ =๐ less than 1. โ f is at continuous for all real numbers less than 1. Case 3 Let x = c (where c > 1) ๐ ๐ฅ๏ทฏ=๐ฅ2 f is continuous at x = c if lim๏ทฎxโ๐๏ทฏ ๐ ๐ฅ๏ทฏ=๐(๐) Thus lim๏ทฎxโ๐๏ทฏ ๐ ๐ฅ๏ทฏ=๐(๐) โ f is continuous at ๐ฅ =๐ (c is greater than 1) โ f is continuous at all real numbers greater than 1. Hence, only x = 1 point of discontinuity โ f is continuous for all real point except 1. Thus, f is continuous for all xโR โ{1}

Ex 5.1