Question

Let $\left[x\right]$ denotes the greatest integer less than or equal to $x$ and $f\left(x\right)=\left[{\tan }^{2}x\right]$. Then.

• A. $\underset{x\to 0}{lim}f\left(x\right)$ does not exist
• B. $f\left(x\right)$ is continuous at $x=0$
• C. $f\left(x\right)$ is not differentiable at $x=0$
• D. $f^{\prime }\left(0\right)=1$

Answer 1

The correct option is B $f\left(x\right)$ is continuous at $x=0$

Given, $f\left(x\right)=\left[{\tan }^{2}x\right]$

We have $-\frac{\pi }{4}<x<\frac{\pi }{4}$
$\Rightarrow -1<\tan x<1$
$\Rightarrow 0\le {\tan }^{2}x<1$
$\Rightarrow \left[{\tan }^{2}x\right]=0$
Therefore, $f\left(x\right)=\left[{\tan }^{2}x\right]=0,\forall xϵ(-\frac{\pi }{4},\frac{\pi }{4})$
Thus, $f\left(x\right)$ is constant function on $(-\frac{\pi }{4},\frac{\pi }{4})$.
Hence, it is continuous on $(-\frac{\pi }{4},\frac{\pi }{4})$ and $f^{\prime }\left(x\right)=0$, $\forall xϵ(-\frac{\pi }{4},\frac{\pi }{4})$.