Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.
Mean Value Theorem states that for a function f:[a,b] -> R, if
(a) f is continuous on [a, b]
(b) f is differentiable on (a, b)
ii) f (x) = [x] for x ∈ [– 2, 2]
It is evident that the given function f (x) is not continuous at every integral point.
In particular, f(x) is not continuous at x = −2 and x = 2
⇒ f (x) is not continuous in [−2, 2].
The differentiability of f in (−2, 2) is checked as follows.
Let n be an integer such that n ∈ (−2, 2).