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Question
Find all points of discontinuity of f, where f is defined by:
f(x) = `{(x^10 - 1", if" x<=1),(x^2", if" x > 1):}`
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Solution
f(x) = `{(x^10 - 1", if" x<=1),(x^2", if" x > 1):}`
For x < 1, f(x) = x10 − 1 and
x > 1, f(x) = x2 is a polynomial function.
So this is a function.
At x = 1,
`lim_(x -> 1^-)` f(x) = `lim_(x -> 1^-)` (x10 − 1)
= `lim_(h -> 0)` [(1 − h)10 − 1]
= (1 − 0)10 − 1
= 1 − 1
= 0
`lim_(x -> 1^+)` f(x) = `lim_(x -> 1^+)` (x2)
= `lim_(h -> 0)` (1 + h)2
= `lim_(h -> 0)` (1 + h2 + 2h)
= 1
Hence, f is not continuous at x = 1.
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