Example 16 - Prove that every rational function is continuous - Examples

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Example 16 Prove that every rational function is continuous. Every Rational number is of the form ﷐𝑝﷮𝑞﷯ where 𝑞 ≠ 0 & 𝑝, 𝑞 are polynomial function Let 𝑓﷐𝑥﷯= ﷐𝑝﷐𝑥﷯﷮𝑞﷐𝑥﷯﷯ , 𝑞﷐𝑥﷯ ≠ 0 Where 𝑝﷐𝑥﷯ & 𝑞﷐𝑥﷯ are polynomials Since 𝑝﷐𝑥﷯ & 𝑞﷐𝑥﷯ are polynomials and we know that every polynomial function is continuous. ⇒ 𝑝﷐𝑥﷯ & 𝑞﷐𝑥﷯ both continuous By Algebra of continuous function if 𝑝, 𝑞 are continuous then ﷐𝑝﷮𝑞﷯ is continuous. ⇒ 𝑓﷐𝑥﷯= ﷐𝑝﷐𝑥﷯﷮𝑞﷐𝑥﷯﷯ is continuous for all real numbers except at points where 𝑞﷐𝑥﷯ ≠ 0

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