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Ex 5.1, 7 - Find all points of discontinuity of f(x) = {|x| + 3 - Ex 5.1

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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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‰คโˆ’3๏ทฎ โˆ’2๐‘ฅ, ๐‘–๐‘“โˆ’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‰คโˆ’3๏ทฎ โˆ’2๐‘ฅ, ๐‘–๐‘“โˆ’3<๐‘ฅ<3๏ทฎ 6๐‘ฅ+2, ๐‘–๐‘“ ๐‘ฅโ‰ฅ3๏ทฏ๏ทฏ Case 1 At ๐‘ฅ =โˆ’3 f is continuous at x = โˆ’ 3 if L.H.L = R.H.L = ๐‘“๏ทโˆ’3๏ทฏ & ๐‘“๏ท๐‘ฅ๏ทฏ = ๏ท๐‘ฅ๏ทฏ+3 ๐‘“๏ทโˆ’3๏ทฏ = ๏ทโˆ’3๏ทฏ + 3 = 3 + 3 = 6 Thus, L.H.L = R.H.L = ๐‘“๏ทโˆ’3๏ทฏ โ‡’ f is continuous at x = โˆ’ 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. โˆ’ 6 โ‰  20 โ‡’ f is not continuous at x = 3 Case 3 Let ๐‘ฅ =๐‘, where โˆ’3 < c < 3 ๐‘“๏ท๐‘ฅ๏ทฏ=โˆ’2๐‘ฅ f is continuous at x = c if ๏ทlim๏ทฎxโ†’๐‘๏ทฏ ๐‘“๏ท๐‘ฅ๏ทฏ=๐‘“(๐‘) โˆด f is continuous at ๐‘ฅ =๐‘ & โˆ’3 < x < 3 Thus, f is continuous at each x โˆˆ(โˆ’3, 3) Case 4 Let ๐‘ฅ =๐‘ where c < โˆ’3) โˆด ๐‘“๏ท๐‘ฅ๏ทฏ= ๏ท๐‘ฅ๏ทฏ+3 f is continuous at x = c (where c < โˆ’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‘<โˆ’3) โ‡’ f is continuous at all real numbers less than โˆ’ 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 โˆˆRโˆ’๏ท3๏ทฏ

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