Example 19 Show that the function defined by f (x) = sin (x2) is a continuous function.Given 𝑓(π‘₯) = sin⁑(π‘₯^2 ) Let π’ˆ(𝒙) = sin⁑π‘₯ & 𝒉(𝒙) = π‘₯^2 Now, (π’ˆ 𝒐 𝒉)(𝒙) = g(β„Ž(π‘₯)) = 𝑔(π‘₯^2 ) = sin⁑(π‘₯^2 ) = 𝒇(𝒙) So, we can write 𝑓(π‘₯) = π‘”π‘œβ„Ž Here, 𝑔(π‘₯) = sin⁑π‘₯ is continuous & β„Ž(π‘₯) = π‘₯^2 is continuous being a polynomial . We know that if two function 𝑔 & β„Ž are continuous then their composition π’ˆπ’π’‰ is continuous Hence, π‘”π‘œβ„Ž(π‘₯) is continuous ∴ 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 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.