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

Transcript

Ex 5.1, 13 Is the function defined by ๐ ๐ฅ๏ทฏ= ๐ฅ+5, ๐๐ ๐ฅโค1๏ทฎ& ๐ฅโ5 , ๐๐ ๐ฅ>1๏ทฏ๏ทฏ a continuous function? Given function is , ๐ ๐ฅ๏ทฏ= ๐ฅ+5, ๐๐ ๐ฅโค1๏ทฎ& ๐ฅโ5 , 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 ๐ ๐ฅ๏ทฏ=๐ฅ+5 f is continuous at x = c if if lim๏ทฎxโ๐๏ทฏ ๐ ๐ฅ๏ทฏ=๐(๐) Thus, lim๏ทฎxโ๐๏ทฏ ๐ ๐ฅ๏ทฏ=๐ ๐๏ทฏ 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ฅโ5 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 for ๐ฅ =๐ ( c is greater than 1) โ f is continuous at all real numbers greater than 1. Hence, only x = 1 is point of discontinuity. โ f is continuous at all real point. Thus, f is continuous for all ๐โ๐โ{๐}.