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

Transcript

Example 11 Find all the points of discontinuity of the function f defined by ๐๏ท๐ฅ๏ทฏ=๏ท๏ท&๐ฅ+2 ,๐๐ ๐ฅ<1๏ทฎ0 , ๐๐ ๐ฅ=1๏ทฎ&๐ฅโ2 ,๐๐ ๐ฅ>1๏ทฏ๏ทฏ Given ๐๏ท๐ฅ๏ทฏ=๏ท๏ท&๐ฅ+2 ,๐๐ ๐ฅ<1๏ทฎ0 , ๐๐ ๐ฅ=1๏ทฎ&๐ฅโ2 ,๐๐ ๐ฅ>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. ๏ทlim๏ทฎxโ๏ท1๏ทฎโ๏ทฏ๏ทฏ ๐๏ท๐ฅ๏ทฏ=๏ทlim๏ทฎxโ๏ท1๏ทฎ+๏ทฏ๏ทฏ ๐๏ท๐ฅ๏ทฏ=๐(1) Since, L.H.L โ  R.H.L โ f is not continuous at ๐ฅ=1. Case 2 Let c be any real number greater than 1. So, ๐ฅ =๐ where c >1 โด ๐๏ท๐ฅ๏ทฏ=๐ฅโ2 f is continuous at x = c if ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ= ๐๏ท๐๏ทฏ Hence, f is continuous at ๐ฅ =๐ (c is greater than 1) โ f is continuous at all point ๐ฅ>1 Case 3 Let c be any real number less than 1. So, ๐ฅ =๐ where c < 1 ๐๏ท๐ฅ๏ทฏ=๐ฅ+2 f is continuous at x = c if ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ= ๐๏ท๐๏ทฏ Hence ๏ทlim๏ทฎxโ๐๏ทฏ ๐๏ท๐ฅ๏ทฏ= ๐๏ท๐๏ทฏ โด f is continuous at for all real number less than 1. Hence only ๐ฅ=1 is point of discontinuty. โ f is continuous at all real point except 1. Thus, f is continuous for ๐ โ๐โ{๐}

Examples