Example 20 - Show that f(x) = |1 - x + |x|| is continuous - Examples

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Example 20 Show that the function f defined by f (x) = |1− 𝑥 + | 𝑥 ||, where x is any real number is a continuous 𝑓﷐𝑥﷯ = ﷐1−𝑥+﷐𝑥﷯﷯ Let 𝑔(𝑥) = 1−𝑥+﷐𝑥﷯ & ℎ(𝑥) = ﷐𝑥﷯ Then ﷐ℎ 𝑜 𝑔﷯﷐𝑥﷯ = ℎ﷐𝑔﷐𝑥﷯﷯ = ℎ﷐1−𝑥+﷐𝑥﷯﷯ = ﷐1−𝑥+﷐𝑥﷯﷯ Now ℎ(𝑥) = ﷐𝑥﷯ We know that Modulus function is continuous ⇒ ℎ(𝑥) = ﷐𝑥﷯ is continuous 𝑔(𝑥) = ﷐1−𝑥﷯+﷐𝑥﷯ Since ﷐1−𝑥﷯ is a polynomial & we know that every polynomial function is continuous ⇒ ﷐1−𝑥﷯ is continuous & ℎ(𝑥) = ﷐𝑥﷯ is also continuous We know that Sum of two continuous function is also continuous 𝑔﷐𝑥﷯ = 1−𝑥+﷐𝑥﷯ is continuous . Hence 𝑔﷐𝑥﷯ & ℎ﷐𝑥﷯ are both continuous . If two function of 𝑔﷐𝑥﷯ & ℎ﷐𝑥﷯ both continuous then their composition ﷐ℎ 𝑜 𝑔﷯﷐𝑥﷯ is also continuous Hence 𝒇﷐𝒙﷯ is continuous .

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