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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):}`
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Solution
We observe that f is continuous at all real numbers x < 1 and x > 1.
Now, continuity at x = 1
`lim_(x -> 1^-)` f(x) = `lim_(x -> 1^-)` (x2 + 1)
= `lim_(h -> 0)` [(1 − h)2 + 1]
= `lim_(h -> 0)` [1 + h2 − 2h + 1]
= `lim_(h -> 0)` [2 + h2 − 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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