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Question
Discuss the continuity and differentiability of f(x) at x = 2
f(x) = [x] if x ∈ [0, 4). [where [*] is a greatest integer (floor) function]
Solution
Explanation:
x ∈ [0, 4)
∴ 0 ≤ x < 4
we will plot graph for 0 ≤ x < 4
Not for x < 0 and upto x = 4 making on X-axis.
f(x) = [x]
∵ Greatest integer function is discontinuous at all integer values of x and hence not differentiable at all integers.
∴ f is not continuous at x = 2.
∵ f(x) `{:(= 1, x < 2),(= 2, x ≥ 2):}}` x ∈ neighbourhood of x = 2.
∴ L.h.lim = 1, R.h.lim = 2
∴ f is not a continuous function.
∴ f is not differentiable function.
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