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Question
Discuss the continuity of the following function at the point(s) or on the interval indicated against them:
f(x) = [x + 1] for x ∈ [−2, 2)
Where [*] is greatest integer function.
Solution
f(x) = [x + 1] ; x ∈ [−2, 2)
∴ f(x) `{:(= -1 , ";" x ∈ [−2"," -1)),(= 0, ";" x ∈ [−1"," 0)),(= 1, ";" x ∈ [0"," 1)),(= 2, ";" x ∈ [1 "," 2)):}`
For continuity at x = – 1
`lim_(x -> -1^-) "f"(x) = lim_(x -> -1^-) [x + 1]`
`lim_(x -> -1^+) "f"(x) = lim_(x -> -1^+) [x + 1]`
∴ `lim_(x -> -1^-) "f"(x) = lim_(x -> -1^+) "f"(x)`
∴ f(x) is discontinuous at x = – 1
Similarly f(x) is discontinuous at
The points x = 0 and x = 1
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