# Example 11 - Chapter 5 Class 12 Continuity and Differentiability

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Example 11 Find all the points of discontinuity of the function f defined by ๐๏ท๐ฅ๏ทฏ=๏ท๏ท&๐ฅ+2 ,๐๐ ๐ฅ<1๏ทฎ0 , ๐๐ ๐ฅ=1๏ทฎ&๐ฅโ2 ,๐๐ ๐ฅ>1๏ทฏ๏ทฏ Given ๐๏ท๐ฅ๏ทฏ=๏ท๏ท&๐ฅ+2 ,๐๐ ๐ฅ<1๏ทฎ0 , ๐๐ ๐ฅ=1๏ทฎ&๐ฅโ2 ,๐๐ ๐ฅ>1๏ทฏ๏ทฏ Case 1 Checking continuity at x = 1 f is continuous at x = 1 if, L.H.L = R.H.L = ๐(1) i.e. ๏ทlim๏ทฎxโ๏ท1๏ทฎโ๏ทฏ๏ทฏ ๐๏ท๐ฅ๏ทฏ=๏ทlim๏ทฎxโ๏ท1๏ทฎ+๏ทฏ๏ทฏ ๐๏ท๐ฅ๏ทฏ=๐(1) Since, L.H.L โ R.H.L โ f is not continuous at ๐ฅ=1. Case 2 Let c be any real number greater than 1. So, ๐ฅ =๐ where c >1 โด ๐๏ท๐ฅ๏ทฏ=๐ฅโ2 f is continuous at x = c if ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ= ๐๏ท๐๏ทฏ Hence, f is continuous at ๐ฅ =๐ (c is greater than 1) โ f is continuous at all point ๐ฅ>1 Case 3 Let c be any real number less than 1. So, ๐ฅ =๐ where c < 1 ๐๏ท๐ฅ๏ทฏ=๐ฅ+2 f is continuous at x = c if ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ= ๐๏ท๐๏ทฏ Hence ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ= ๐๏ท๐๏ทฏ โด f is continuous at for all real number less than 1. Hence only ๐ฅ=1 is point of discontinuty. โ f is continuous at all real point except 1. Thus, f is continuous for ๐ โ๐โ{๐}

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About the Author

Davneet Singh

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.