Consider the greatest integer function, defined by f ( x ) = [ x ]...

Consider the greatest integer function, defined by

f ( x ) = [ x ] , 0 \le x < 2

. Then

Options

A $f$ is derivable at $x = 1$
B $f$ is not derivable at $x = 1$
C $f$ is derivable at $x = 2$
DNone of these

Correct answer

B. $f$ is not derivable at $x = 1$

Step-by-step solution

The given function is

$f\left(x\right) = \left\{$ \begin{cases}0, & 0 \le x < 1 \\ 1, & 1 \le x < 2\end{cases} $\right)Now,$ LHD = \lim_{x \to {1}^{-}}\dfrac{f ( x ) - f ( 1 )}{x - 1} = \lim_{x \to {1}^{-}}\dfrac{0 - 1}{x - 1} $Which does not exist $ ∴ f $is not derivable at$ x = 1$.

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