Ex 5.1, 34 - Find all points of discontinuity f(x) = |x| - |x+1| - Algebra of continous functions

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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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‘ฅโˆ’ ๐‘ฅ+1๏ทฏ โˆ’x โˆ’ x + 1 = x โˆ’ x โˆ’ 1 = ๐‘ฅโˆ’๐‘ฅ+1 = x โˆ’ x โˆ’ 1 = โˆ’1 When ๐‘ฅ<โˆ’1 ๐‘“ ๐‘ฅ๏ทฏ = โˆ’๐‘ฅโˆ’ โˆ’ ๐‘ฅ+1๏ทฏ๏ทฏ = โˆ’๐‘ฅ+๐‘ฅ+1 = 1 When โˆ’1โ‰ค๐‘ฅ<0 ๐‘“ ๐‘ฅ๏ทฏ = โˆ’๐‘ฅโˆ’ ๐‘ฅ+1๏ทฏ = โˆ’๐‘ฅโˆ’๐‘ฅ+1 = โˆ’2๐‘ฅ+1 Now ๐‘“ ๐‘ฅ๏ทฏ= 1 ๐‘–๐‘“ ๐‘ฅโ‰คโˆ’1๏ทฎโˆ’2๐‘ฅโˆ’1 ๐‘–๐‘“ โˆ’1โ‰ค๐‘ฅ<0๏ทฎโˆ’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๏ทฎโˆ’๏ทฏ๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ= lim๏ทฎxโ†’ 0๏ทฎ+๏ทฏ๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ= ๐‘“ 0๏ทฏ & ๐‘“ ๐‘ฅ๏ทฏ = โˆ’1 ๐‘“ 0๏ทฏ = โˆ’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 = โˆ’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 = โˆ’ 1 A function is continuous at x = โˆ’1 if L.H.L = R.H.L = ๐‘“ โˆ’1๏ทฏ & ๐‘“ ๐‘ฅ๏ทฏ = โˆ’2๐‘ฅโˆ’1 ๐‘“ โˆ’1๏ทฏ = โˆ’2 โˆ’1๏ทฏโˆ’1 = 2 โ€“ 1 = 1 Thus L.H.L = R.H.L = ๐‘“ โˆ’1๏ทฏ Hence ๐’‡ ๐’™๏ทฏ is continuous at x = โˆ’1 Case 4 At x < โˆ’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 < โˆ’1 Case 5 At โˆ’1โ‰ค๐‘ฅ<0 ๐‘“ ๐‘ฅ๏ทฏ = โˆ’2๐‘ฅโˆ’1, is a polynomial & we know that every polynomial function is continuous for all real values . Hence ๐’‡ ๐’™๏ทฏ = โˆ’๐Ÿ๐’™โˆ’๐Ÿ is continuous at โˆ’๐Ÿโ‰ค๐’™<๐ŸŽ โ‡’ There is no point of discontinuity Hence ๐’‡ ๐’™๏ทฏ is continuous for all ๐’™โˆˆ๐‘น

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.