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Question
Is the function defined by f(x) = `{(x+5", if" x <= 1),(x -5", if" x > 1):}` a continuous function?
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Solution
f(x) = `{(x+5", if" x <= 1),(x -5", if" x > 1):}`
`lim_(x -> 1^-)` f(x) = `lim_(x -> 1^-)` (x + 5)
= `lim_(h -> 0)` [1 − h + 5]
= `lim_(h -> 0)` (6 − h)
= 6 − 0
= 6
`lim_(x -> 1^+)` f(x) = `lim_(x -> 1^+)` (x − 5)
= `lim_(h -> 0)` (1 + h − 5)
= `lim_(h -> 0)` (h − 4)
= 0 − 4
= −4
Hence, f is not continuous at x = 1.
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