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

Transcript

Ex 5.1, 7 Find all points of discontinuity of f, where f is defined by ๐๏ท๐ฅ๏ทฏ=๏ท๏ท๏ท๐ฅ๏ทฏ+3, ๐๐ ๐ฅโคโ3๏ทฎ โ2๐ฅ, ๐๐โ3<๐ฅ<3๏ทฎ 6๐ฅ+2, ๐๐ ๐ฅโฅ3๏ทฏ๏ทฏ We have, ๐๏ท๐ฅ๏ทฏ=๏ท ๏ท ๏ท๐ฅ๏ทฏ+3, ๐๐ ๐ฅโคโ3๏ทฎ โ2๐ฅ, ๐๐โ3<๐ฅ<3๏ทฎ 6๐ฅ+2, ๐๐ ๐ฅโฅ3๏ทฏ๏ทฏ Case 1 At ๐ฅ =โ3 f is continuous at x = โ 3 if L.H.L = R.H.L = ๐๏ทโ3๏ทฏ & ๐๏ท๐ฅ๏ทฏ = ๏ท๐ฅ๏ทฏ+3 ๐๏ทโ3๏ทฏ = ๏ทโ3๏ทฏ + 3 = 3 + 3 = 6 Thus, L.H.L = R.H.L = ๐๏ทโ3๏ทฏ โ f is continuous at x = โ 3 Case 2 At ๐ฅ =3 f is continuous at x = 3 if L.H.L = R.H.L = ๐๏ท3๏ทฏ Since L.H.L โ  R.H.L i.e. โ 6 โ  20 โ f is not continuous at x = 3 Case 3 Let ๐ฅ =๐, where โ3 < c < 3 ๐๏ท๐ฅ๏ทฏ=โ2๐ฅ f is continuous at x = c if ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ=๐(๐) โด f is continuous at ๐ฅ =๐ & โ3 < x < 3 Thus, f is continuous at each x โ(โ3, 3) Case 4 Let ๐ฅ =๐ where c < โ3) โด ๐๏ท๐ฅ๏ทฏ= ๏ท๐ฅ๏ทฏ+3 f is continuous at x = c (where c < โ3) if ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ=๐(๐) Hence ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ=๐(๐) โ f is continuous at ๐ฅ =๐ (๐<โ3) โ f is continuous at all real numbers less than โ 3. Case 5 If ๐ฅ =๐ & ๐>3 ๐๏ท๐ฅ๏ทฏ= 6x+2 f is continuous at x = c where c > 3 if ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ=๐(๐) Thus ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ=๐(๐) โ f is continuous at ๐ฅ =๐ (๐>3) โ f is continuous at all real numbers greater than 3. Hence, f is discontinuous at only ๐ฅ =3 โ f is continuous at all real numbers except 3. f is continuous at x โRโ๏ท3๏ทฏ