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

Transcript

Ex 5.1, 16 Discuss the continuity of the function f, where f is defined by ๐ ๐ฅ๏ทฏ= โ&2, ๐๐ ๐ฅโคโ1๏ทฎ2&๐ฅ, ๐๐ โ1โค๐ฅโค1๏ทฎ2, ๐๐ ๐ฅ>1 ๏ทฏ๏ทฏ Case 1:- At x = โ1 A function is continuous at x = โ 1 if L.H.L = R.H.L = ๐ โ1๏ทฏ i.e. lim๏ทฎxโ โ1๏ทฎโ๏ทฏ๏ทฏ ๐ ๐ฅ๏ทฏ = lim๏ทฎxโ โ1๏ทฎ+๏ทฏ๏ทฏ ๐ ๐ฅ๏ทฏ = ๐ โ1๏ทฏ And ๐ โ2๏ทฏ = โ2 Thus L.H.L = R.H.L = ๐ โ2๏ทฏ = โ2 Hence ๐ ๐๏ทฏ is continuous at x = โ๐ Case 1:- At x = โ1 A function is continuous at x = โ 1 if A function is continuous at x = 1 if if L.H.L = R.H.L = ๐ 1๏ทฏ i.e. lim๏ทฎxโ 1๏ทฎโ๏ทฏ๏ทฏ ๐ ๐ฅ๏ทฏ = lim๏ทฎxโ 1๏ทฎ+๏ทฏ๏ทฏ ๐ ๐ฅ๏ทฏ = ๐ 1๏ทฏ & ๐ 1๏ทฏ = 2 1๏ทฏ = 2 Thus L.H.L = R.H.L = ๐ โ2๏ทฏ Hence ๐ ๐๏ทฏ is continuous at x = 1 Case 3:- For ๐ฅ<โ1 ๐ ๐ฅ๏ทฏ = โ2 Thus, ๐ ๐ฅ๏ทฏ is a constant function . & Every constant function is continuous for all real number. Hence ๐ ๐๏ทฏ is continuous at ๐<โ๐ Case 4:- For ๐ฅ>1 ๐ ๐ฅ๏ทฏ = 2 Thus, ๐ ๐ฅ๏ทฏ is a constant function . & Every constant function is continuous for all real number. Hence ๐ ๐๏ทฏ is continuous at ๐>๐ Case 5:- For โ1โค๐ฅโค1 ๐ ๐ฅ๏ทฏ = 2๐ฅ So, f(x) is a polynomial & Every polynomial is continuous. โ ๐ ๐๏ทฏ is continuous at โ๐<๐โค๐ Thus, f(x) is continuous for all real numbers, i.e. f is continuous for all x โ R