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Let $[.]$ denote the greatest integer function and $⨍ (x)= [\tan^{2} x]$ then

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$⨍^{1} (0) = 1$

$⨍ (x)$ is not differentiable at $x = 0$

$⨍^{1} (0)=0$

$⨍ (x)$ is not continuous at $x = 0$

answer is D.

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Detailed Solution

$0 \leq \tan^{2} x < 1 \Leftrightarrow \frac{- π}{4} < x < \frac{π}{4} f (x) = [\tan^{2} x] = 0 if x \in (\frac{- π}{4}, \frac{π}{4})$

$∴ f (x)$ is continuous and differentiable at $x = 0 and f^{1} (0) = 0$

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