Check Full Chapter Explained - Continuity and Differentiability - https://you.tube/Chapter-5-Class-12-Continuity

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

Transcript

Example 20 Show that the function f defined by f (x) = |1โˆ’ ๐‘ฅ + | ๐‘ฅ ||, where x is any real number is a continuous ๐‘“(๐‘ฅ) = |(1โˆ’๐‘ฅ+|๐‘ฅ|)| Let ๐‘”(๐‘ฅ) = 1โˆ’๐‘ฅ+|๐‘ฅ| & โ„Ž(๐‘ฅ) = |๐‘ฅ| Then , โ„Ž๐‘œ๐‘”(๐‘ฅ) = โ„Ž(๐‘”(๐‘ฅ)) = โ„Ž(1โˆ’๐‘ฅ+|๐‘ฅ|) = |(1โˆ’๐‘ฅ+|๐‘ฅ|)| Now โ„Ž(๐‘ฅ) = |๐‘ฅ| We know that Modulus function is continuous โˆด โ„Ž(๐‘ฅ) = |๐‘ฅ| is continuous ๐‘”(๐‘ฅ) = (1โˆ’๐‘ฅ)+|๐‘ฅ| Since (1โˆ’๐‘ฅ) is a polynomial & every polynomial function is continuous โˆด (1โˆ’๐‘ฅ) is continuous Also, |๐‘ฅ| is also continuous So, 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 . If two function of ๐‘”(๐‘ฅ) & โ„Ž(๐‘ฅ) both continuous, then their composition โ„Ž๐‘œ๐‘”(๐‘ฅ) 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.