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Question
Find all points of discontinuity of f, where f is defined by:
f(x) = `{(2x + 3", if" x<=2),(2x - 3", if" x > 2):}`
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Solution
f(x) = `{(2x + 3", if" x<=2),(2x - 3", if" x > 2):}`
`lim_(x -> 2^-)` f(x) = `lim_(x -> 2^-)` (2x + 3)
= `lim_(h -> 0)` [2(2 − h) + 3]
= `lim_(h -> 0)` [4 − 2h + 3]
= `lim_(h -> 0)` (7 − 2h)
= 7 − 2 × 0
= 7
`lim_(x -> 2^+)` f(x) = `lim_(x -> 2^+)` (2x − 3)
= `lim_(h -> 0)` [2(2 + h) − 3]
= `lim_(h ->0)` [4 + 2h − 3]
= `lim_(h ->0)` (1 + 2h)
= 1 + 2 × 0
= 1
Therefore, f is not continuous at x = 2.
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