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Example 14 - Show that every polynomial function is continuous

Example 14 - Chapter 5 Class 12 Continuity and Differentiability - Part 2

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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) (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 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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