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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):}`
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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