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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Example 20 Show that the function f defined by f (x) = |1− 𝑥 + | 𝑥 ||, where x is any real number is a continuousGiven 𝑓(𝑥) = |(1−𝑥+|𝑥|)| Let 𝒈(𝒙) = 1−𝑥+|𝑥| & 𝒉(𝒙) = |𝑥| Then , 𝒉𝒐𝒈(𝒙) = ℎ(𝑔(𝑥)) = ℎ(1−𝑥+|𝑥|) = |(1−𝑥+|𝑥|)| = 𝒇(𝒙) We know that, Modulus function is continuous ∴ 𝒉(𝒙) = |𝑥| is continuous Also, 𝒈(𝒙) = (𝟏−𝒙)+|𝒙| Since (1−𝑥) is a polynomial & every polynomial function is continuous ∴ (𝟏−𝒙) is continuous Also, |𝒙| is also continuous Since Sum of two continuous function is also continuous Thus, 𝑔(𝑥) = 1−𝑥+|𝑥| is continuous . Hence, 𝑔(𝑥) & ℎ(𝑥) are both continuous . We know that If two function of 𝑔(𝑥) & ℎ(𝑥) both continuous, then their composition 𝒉𝒐𝒈(𝒙) is also continuous Hence, 𝒇(𝒙) is continuous .

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.