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]

Answer 1

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.