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Example 12 - Discuss continuity of f(x) = {x + 2, -x + 2 - Examples

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Example 12 Discuss the continuity of the function defined by ๐‘“๏ท๐‘ฅ๏ทฏ=๏ท๏ท& ๐‘ฅ+2, ๐‘–๐‘“ ๐‘ฅ<0๏ทฎ&โˆ’๐‘ฅ+2, ๐‘–๐‘“ ๐‘ฅ>0๏ทฏ๏ทฏ We have, ๐‘“๏ท๐‘ฅ๏ทฏ=๏ท๏ท& ๐‘ฅ+2, ๐‘–๐‘“ ๐‘ฅ<0๏ทฎ&โˆ’๐‘ฅ+2, ๐‘–๐‘“ ๐‘ฅ>0๏ทฏ๏ทฏ Here, the given function is not defined at ๐‘ฅ=0. So, we check continuity for all real numbers except 0. Case 1 Let ๐‘ฅ=๐‘ where ๐‘<0. ๐‘“๏ท๐‘ฅ๏ทฏ=๐‘ฅ+2 ๐‘“ is continuous at ๐‘ฅ=๐‘ 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‡’ ๐‘“ is continuous at ๐‘ฅ=๐‘ where ๐‘<0. โ‡’ ๐‘“ is continuous at all Real points less than 0. Case 2 Let ๐‘ฅ=๐‘ where ๐‘<0. ๐‘“๏ท๐‘ฅ๏ทฏ=โˆ’๐‘ฅ+2 ๐‘“ is continuous at ๐‘ฅ=๐‘ 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‡’ ๐‘“ is continuous at ๐‘ฅ=๐‘ where ๐‘>0. โ‡’ ๐‘“ is continuous for all Real points greater than 0. Hence, ๐‘“ is continuous for all Real points except 0. โ‡’ ๐’‡ is continuous for ๐’™ โˆˆ๐‘โˆ’๏ท๐ŸŽ๏ทฏ

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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