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Question
Is the function f defined by f(x) = `{(x", if" x<=1),(5", if" x > 1):}` continuous at x = 0? At x = 1? At x = 2?
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Solution
f(x) = `{(x", if" x<=1),(5", if" x > 1):}`
(i) At x = 0
`lim_(x -> 0^-)` f(x) = `lim_(h -> 0)` f(0 − h)
= 0 − 0
= 0
`lim_(x -> 0^+)` f(x) = `lim_(h -> 0)` f(0 + h)
= 0 + 0
= 0
f(0) = 0
Hence, f is continuous at x = 0.
(ii) At x = 1
`lim_(x -> 1^-)` f(x) = `lim_(h -> 0)` f(1 − h)
= 1 − 0
= 1
`lim_(x -> 1^+)` f(x) = `lim_(h -> 0)` f(1 + h)
= 5
f(1) = 1
Hence, f is not continuous at x = 1.
(iii) At x = 2
`lim_(x -> 2^-)` f(x) = `lim_(h -> 0)` f(2 − h)
= 5
`lim_(x -> 2^+)` f(x) = `lim_(h -> 0)` f(2 + h)
= 5
f(2) = 5
Hence, f is continuous at x = 2.
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