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Example 20 - Show that function f(x) = |1 - x + |x|| is continous

Example 20 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Example 20 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

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Example 20 Show that the function f defined by f (x) = |1βˆ’ π‘₯ + | π‘₯ ||, where x is any real number is a continuousGiven 𝑓(π‘₯) = |(1βˆ’π‘₯+|π‘₯|)| Let π’ˆ(𝒙) = 1βˆ’π‘₯+|π‘₯| & 𝒉(𝒙) = |π‘₯| Then , π’‰π’π’ˆ(𝒙) = β„Ž(𝑔(π‘₯)) = β„Ž(1βˆ’π‘₯+|π‘₯|) = |(1βˆ’π‘₯+|π‘₯|)| = 𝒇(𝒙) We know that, Modulus function is continuous ∴ 𝒉(𝒙) = |π‘₯| is continuous Also, π’ˆ(𝒙) = (πŸβˆ’π’™)+|𝒙| Since (1βˆ’π‘₯) is a polynomial & every polynomial function is continuous ∴ (πŸβˆ’π’™) is continuous Also, |𝒙| is also continuous Since Sum of two continuous function is also continuous Thus, 𝑔(π‘₯) = 1βˆ’π‘₯+|π‘₯| is continuous . Hence, 𝑔(π‘₯) & β„Ž(π‘₯) are both continuous . We know that If two function of 𝑔(π‘₯) & β„Ž(π‘₯) both continuous, then their composition π’‰π’π’ˆ(𝒙) is also continuous Hence, 𝒇(𝒙) is continuous .

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.