Question

$\left[{\tan }^{2}x\right]$ is continuous and differentiable at $x=0$ (where $[\cdot ]$ denotes greatest integer).If this is true enter 1,If false enter 0.

Answer 1

$\underset{h\to 0}{lim}f(0+h)=\underset{h\to 0}{lim}\left[ta{n}^{2}h\right]=0$
$\underset{h\to 0}{lim}f(0-h)=\underset{h\to 0}{lim}\left[ta{n}^2\right(-h\left)\right]=0$
Hence $f\left(x\right)$ is continuous at $x=0.$
$f^{\prime }\left(0^{+}\right)=\underset{h\to 0}{lim}\frac{f\left(h\right)-f\left(0\right)}{h}=0$
$f^{\prime }\left(0^{-}\right)=\underset{h\to 0}{lim}\frac{f(-h)-f\left(0\right)}{-h}=0$
Hence $f\left(x\right)$ is differentiable at $x=0.$