Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12



  1. Chapter 5 Class 12 Continuity and Differentiability
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Example 20 Show that the function f defined by f (x) = |1โˆ’ ๐‘ฅ + | ๐‘ฅ ||, where x is any real number is a continuousGiven ๐‘“(๐‘ฅ) = |(1โˆ’๐‘ฅ+|๐‘ฅ|)| Let ๐’ˆ(๐’™) = 1โˆ’๐‘ฅ+|๐‘ฅ| & ๐’‰(๐’™) = |๐‘ฅ| Then , ๐’‰๐’๐’ˆ(๐’™) = โ„Ž(๐‘”(๐‘ฅ)) = โ„Ž(1โˆ’๐‘ฅ+|๐‘ฅ|) = |(1โˆ’๐‘ฅ+|๐‘ฅ|)| = ๐’‡(๐’™) We know that, Modulus function is continuous โˆด ๐’‰(๐’™) = |๐‘ฅ| is continuous Also, ๐’ˆ(๐’™) = (๐Ÿโˆ’๐’™)+|๐’™| Since (1โˆ’๐‘ฅ) is a polynomial & every polynomial function is continuous โˆด (๐Ÿโˆ’๐’™) is continuous Also, |๐’™| is also continuous Since Sum of two continuous function is also continuous Thus, ๐‘”(๐‘ฅ) = 1โˆ’๐‘ฅ+|๐‘ฅ| is continuous . Hence, ๐‘”(๐‘ฅ) & โ„Ž(๐‘ฅ) are both continuous . We know that If two function of ๐‘”(๐‘ฅ) & โ„Ž(๐‘ฅ) both continuous, then their composition ๐’‰๐’๐’ˆ(๐’™) is also continuous Hence, ๐’‡(๐’™) is continuous .

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.