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Question
Find all point of discontinuity of f, where f is defined by `f (x) = {(2x + 3, if x<=2),(2x - 3, if x > 2):}`
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 xx 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 xx 0`
= 1
Therefore, f is not continuous at x = 2.
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