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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 ?
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Solution
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.
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