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Question
Find all points of discontinuity of f, where f is defined by:
f(x) = `{(|x|+3", if" x<= -3),(-2x", if" -3 < x < 3),(6x + 2", if" x >= 3):}`
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Solution
f(x) = `{(|x|+3", if" x<= -3),(-2x", if" -3 < x < 3),(6x + 2", if" x >= 3):}`
⇒ At x = −3
`lim_(x -> 3^-)` f(x) = `lim_(x -> 3^-)` (|x| + 3)
= `lim_(h -> 0)` [|−3 − h| + 3]
= `lim_(h -> 0)` (6 + h)
= 6 + 0
= 6
`lim_(x -> 3^+)` f(x) = `lim_(x -> 3^+)` (−2x)
= `lim_(h -> 0)` [−2 (−3 + h)]
= `lim_(h -> 0)` (6 − 2h)
= 6 − 2 × 0
= 6
Hence, f is continuous at x = −3.
⇒ At x = 3
`lim_(x -> 3^-)` f(x) = `lim_(x -> 3^-)` (−2x)
= `lim_(h -> 0)` [−2 (3 − h)]
= `lim_(h -> 0)` (−6 + 2h)
= −6 + 2 × 0
= −6
`lim_(x -> 3^+)` f(x) = `lim_(x -> 3^+)` (6x + 2)
= `lim_(h -> 0)` [6(3 + h) + 2]
= `lim_(h -> 0)` (18 + 6h + 2)
= `lim_(h -> 0)` (20 + 6h)
= 20 + 6 × 0
= 20
Hence, f is not continuous at x = 3.
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