Let [.] denote the greatest integer function and $f (x) = [\tan^2x]$. Then,
Answer & explanation
Correct answer: option 2
We have $f (0) = [\tan^2 0] = [0] = 0$,
$f(0+0)=\underset{x→0}{\lim}f(x)=\underset{h→0}{\lim}[\tan^2(0+h)]$, where h is positive and sufficiently small
$=\underset{h→0}{\lim}[\tan^2h]=\underset{h→0}{\lim}0=0$
and $f(0+0)=\underset{x→0^-}{\lim}f(x)=\underset{h→0}{\lim}[\tan^2(0-h)]$, where h is positive and sufficiently small
$=\underset{h→0}{\lim}[\tan^2h]=\underset{h→0}{\lim}0=0$
Since f(0 − 0) = f(0) = f(0 + 0), therefore f(x) is continuous at x = 0.