Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x = [x] for – 1 ≤ x ≤ 1, where [x] denotes the greatest integer not exceeding x
Given function is:
⇒ f(x) = [x], – 1≤x≤1 where [x] denotes the greatest integer not exceeding x.
Let us check the continuity of the function ‘f’.
Here in the interval xϵ[ – 1,1], the function has to be Right continuous at x = 1 and left continuous at x = 1.
⇒ 
⇒ 
where h>0.
⇒ 
⇒ 
......(1)
⇒ 
⇒ 
, where h>0
⇒ 
⇒ 
......(2)
From (1) and (2), we can see that the limits are not the same so, the function is not continuous in the interval [ – 1,1].
∴ Rolle’s theorem is not applicable for the function f in the interval [ – 1,1].
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