Example 14 Show that every polynomial function is continuousLet 𝒇(𝒙)=𝒂_𝟎+𝒂_𝟏 𝒙+𝒂_𝟏 𝒙^𝟐+ … +𝒂_𝒏 𝒙^𝒏 π‘›βˆˆπ’ be a polynomial function Since Polynomial function is valid for every real number We prove continuity of Polynomial Function at any point c Let c be any real number f(x) is continuous at π‘₯ = 𝑐 if (π₯𝐒𝐦)┬(𝐱→𝒄) 𝒇(𝒙)= 𝒇(𝒄) L.H.S (π₯𝐒𝐦)┬(𝐱→𝒄) 𝒇(𝒙) = lim┬(x→𝑐) " " (π‘Ž_0+π‘Ž_1 π‘₯+ … +π‘Ž_𝑛 π‘₯^𝑛 ) Putting x = c = π‘Ž_0+π‘Ž_1 𝑐+ … +π‘Ž_𝑛 𝑐^𝑛 R.H.S 𝒇(𝒄) = π‘Ž_0+π‘Ž_1 𝑐+ … +π‘Ž_𝑛 𝑐^𝑛 Since, L.H.S = R.H.S ∴ Function is continuous at x = c Thus, we can write that f is continuous for all 𝒙 βˆˆπ‘ i.e. Every polynomial function 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.