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

Transcript

Ex 5.1, 34 Find all the points of discontinuity of f defined by ๐ ๐ฅ๏ทฏ= ๐ฅ๏ทฏ โ ๐ฅ+1๏ทฏ. When ๐ฅโฅ0 ๐ ๐ฅ๏ทฏ = ๐ฅโ ๐ฅ+1๏ทฏ โx โ x + 1 = x โ x โ 1 = ๐ฅโ๐ฅ+1 = x โ x โ 1 = โ1 When ๐ฅ<โ1 ๐ ๐ฅ๏ทฏ = โ๐ฅโ โ ๐ฅ+1๏ทฏ๏ทฏ = โ๐ฅ+๐ฅ+1 = 1 When โ1โค๐ฅ<0 ๐ ๐ฅ๏ทฏ = โ๐ฅโ ๐ฅ+1๏ทฏ = โ๐ฅโ๐ฅ+1 = โ2๐ฅ+1 Now ๐ ๐ฅ๏ทฏ= 1 ๐๐ ๐ฅโคโ1๏ทฎโ2๐ฅโ1 ๐๐ โ1โค๐ฅ<0๏ทฎโ1 ๐๐ ๐ฅโฅ0๏ทฏ๏ทฏ Checking continuity Case 1 At x = 0 A function is continuous at x = 0 if L.H.L = R.H.L = ๐ 0๏ทฏ i.e. lim๏ทฎxโ 0๏ทฎโ๏ทฏ๏ทฏ ๐ ๐ฅ๏ทฏ= lim๏ทฎxโ 0๏ทฎ+๏ทฏ๏ทฏ ๐ ๐ฅ๏ทฏ= ๐ 0๏ทฏ & ๐ ๐ฅ๏ทฏ = โ1 ๐ 0๏ทฏ = โ1 Thus L.H.L = R.H.L = ๐ 0๏ทฏ Hence ๐ ๐๏ทฏ is continuous at x = 0 Case 2 At x > 0 ๐ ๐ฅ๏ทฏ = โ1 it is a constant function. & Every constant function is continuous for all real number. โ ๐ ๐๏ทฏ is continuous for x > 0 . Case 3 At x = โ 1 A function is continuous at x = โ1 if L.H.L = R.H.L = ๐ โ1๏ทฏ & ๐ ๐ฅ๏ทฏ = โ2๐ฅโ1 ๐ โ1๏ทฏ = โ2 โ1๏ทฏโ1 = 2 โ 1 = 1 Thus L.H.L = R.H.L = ๐ โ1๏ทฏ Hence ๐ ๐๏ทฏ is continuous at x = โ1 Case 4 At x < โ1 ๐ ๐ฅ๏ทฏ = 1 , it is a constant function & Every constant function is continuous for all real number โ ๐ ๐๏ทฏ is continuous for x < โ1 Case 5 At โ1โค๐ฅ<0 ๐ ๐ฅ๏ทฏ = โ2๐ฅโ1, is a polynomial & we know that every polynomial function is continuous for all real values . Hence ๐ ๐๏ทฏ = โ๐๐โ๐ is continuous at โ๐โค๐<๐ โ There is no point of discontinuity Hence ๐ ๐๏ทฏ is continuous for all ๐โ๐น