Ex 5.1, 13 - Is f(x) = {x+5, if x<1 x-5, if x>1 continuous - Checking continuity using LHL and RHL

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Ex 5.1, 13 Is the function defined by ๐‘“ ๐‘ฅ๏ทฏ= ๐‘ฅ+5, ๐‘–๐‘“ ๐‘ฅโ‰ค1๏ทฎ& ๐‘ฅโˆ’5 , ๐‘–๐‘“ ๐‘ฅ>1๏ทฏ๏ทฏ a continuous function? Given function is , ๐‘“ ๐‘ฅ๏ทฏ= ๐‘ฅ+5, ๐‘–๐‘“ ๐‘ฅโ‰ค1๏ทฎ& ๐‘ฅโˆ’5 , ๐‘–๐‘“ ๐‘ฅ>1๏ทฏ๏ทฏ Case 1 At x = 1 f is continuous at x = 1 if L.H.L = R.H.L = ๐‘“ 1๏ทฏ i.e. if lim๏ทฎxโ†’ 1๏ทฎโˆ’๏ทฏ๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ = lim๏ทฎxโ†’ 1๏ทฎ+๏ทฏ๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ = ๐‘“ 1๏ทฏ Thus, L.H.L โ‰  R.H.L โ‡’ f is discontinuous at ๐’™ =๐Ÿ Case 2 Let x = c , where c < 1 ๐‘“ ๐‘ฅ๏ทฏ=๐‘ฅ+5 f is continuous at x = c if if lim๏ทฎxโ†’๐‘๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ=๐‘“(๐‘) Thus, lim๏ทฎxโ†’๐‘๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ=๐‘“ ๐‘๏ทฏ โ‡’ f is continuous for ๐‘ฅ =๐‘ less than 1. โ‡’ f is at continuous for all real numbers less than 1. Case 3 Let x = c (where c > 1) ๐‘“ ๐‘ฅ๏ทฏ= ๐‘ฅโˆ’5 f is continuous at x = c if lim๏ทฎxโ†’๐‘๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ=๐‘“(๐‘) Thus lim๏ทฎxโ†’๐‘๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ=๐‘“(๐‘) โ‡’ f is continuous for ๐‘ฅ =๐‘ ( c is greater than 1) โ‡’ f is continuous at all real numbers greater than 1. Hence, only x = 1 is point of discontinuity. โ‡’ f is continuous at all real point. Thus, f is continuous for all ๐’™โˆˆ๐‘โˆ’{๐Ÿ}.

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