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Question
Find all points of discontinuity of f, where f is defined by `f (x) = {(x+1, if x>=1),(x^2+1, if x < 1):}`
Solution
`f (x) = {(x+1, if x>=1),(x^2+1, if x < 1):}`
For x > 1, f(x) = x + 1 and
x < 1, f(x) = x2 + 1 is a polynomial function
So this is the function.
⇒ At x = 1,
`lim_(x -> 1^-)` f(x) = `lim_(x -> 1^-)` (x2 + 1)
`= lim_(h -> 0) [(1 - h)^2 + 1]`
`= lim_(h -> 0) [1 + h^2 - 2h + 1]`
`= lim_(h -> 0) [2 + h^2 - 2h]`
= 2 + 0 - 0
= 2
`lim_(x -> 1^+)` f(x) = `lim_(x -> 1^+)` (x + 1)
= `lim_(h -> 0)` (1 + h + 1)
= `lim_(h -> 0)` (2 + h) = 2 + 0 = 2
f(1) = 1 + 1 = 2
Hence, f is a function at x = 1.
There are no points of discontinuity here.
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