Example 7 - Is f(x) = |x| a continuous function - Class 12 - Checking continuity using LHL and RHL

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Example 7 Is the function defined by f (x) = |x|, a continuous function? 𝑓﷐𝑥﷯=﷐𝑥﷯ 𝑓﷐𝑥﷯ = ﷐﷐−𝑥 𝑥<0﷮𝑥 𝑥≥0﷯﷯ Checking continuity Case 1: At x = 0 f is continuous at x = 0 if, L.H.L = R.H.L = 𝑓﷐0﷯ i.e. ﷐lim﷮x→﷐0﷮−﷯﷯ 𝑓﷐𝑥﷯=﷐lim﷮x→﷐0﷮+﷯﷯ 𝑓﷐𝑥﷯=𝑓﷐0﷯ & 𝑓﷐0﷯ = 0 Hence L.H.L = R.H.L = 𝑓﷐0﷯ ∴ f(𝑥) is continuous at x = 0 Case 2 At x = c , c < 0 𝑓﷐𝑥﷯ = −𝑥 𝑓 is continuous at x = c if ﷐lim﷮x→𝑐﷯ 𝑓﷐𝑥﷯=𝑓﷐𝑐﷯ Hence ﷐lim﷮x→𝑐﷯ 𝑓﷐𝑥﷯=𝑓﷐𝑐﷯ therefore f is continuous at x = c (c < 0) ⇒ f is continuous for all real number less then 0. Case 3 At x = c , c > 0 𝑓﷐𝑥﷯ = −𝑥 𝑓 is continuous at x = c if ﷐lim﷮x→𝑐﷯ 𝑓﷐𝑥﷯=𝑓﷐𝑐﷯ Hence ﷐lim﷮x→𝑐﷯ 𝑓﷐𝑥﷯=𝑓﷐𝑐﷯ ∴ f is continuous at x = c (c > 0) ⇒ f is continuous for all real number greater then 0. Thus f is continuous for all real number

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