Question

Let [] denote the greatest integer function and $f\left(x\right)=\left[{\tan }^{2}x\right]$. Then

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

Answer 1

The correct option is B f(x) is continuous at x=0

$f\left(x\right)=\left[{\tan }^{2}x\right]=1$ for $-\pi /4<x<\pi /4$.
Thus $\underset{x\to 0}{lim}f\left(x\right)$ exists and the value is 0.
Moreover, it is continuous at $x=0$.
Being a constant function $f$ is differentiable at $x=0$ and $f^{\prime }\left(0\right)=0.$