Question

f (x) = [x] for −1 ≤ x ≤ 1, where [x] denotes the greatest integer not exceeding x Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?

Answer 1

The given function is \[f\left( x \right) = \left[ x \right]\] .

The domain of f is given to be  \[\left[ - 1, 1 \right]\].

Let \[c \in \left[ - 1, 1 \right]\] such that c is not an integer.

Then, \[\lim_{x \to c} f\left( x \right) = f\left( c \right)\]

Thus, \[f\left( x \right)\]  is continuous at \[x = c\].

Now, let  \[c = 0\]. 

Then,

\[\lim_{x \to 0^-} f\left( x \right) = - 1 \neq 0 = f\left( 0 \right)\]

Thus, f  is discontinuous at x = 0.

Therefore, \[f\left( x \right)\]  is not continuous in  \[\left[ - 1, 1 \right]\] .

Rolle's theorem is not applicable for the given function.