Question
Let [.] denote the greatest integer function and $$f\left( x \right) = \left[ {{{\tan }^2}x} \right],$$ then:
A.
$$\mathop {\lim }\limits_{x \to 0} f\left( x \right)$$ 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.
$$f'\left( 0 \right) = 1$$
Answer :
$$f\left( x \right)$$ is continuous at $$x = 0$$
Solution :
We have $$f\left( x \right) = \left[ {{{\tan }^2}x} \right]$$
$$\tan \,x$$ is an increasing function for $$ - \frac{\pi }{4} < x < \frac{\pi }{4}$$
$$\eqalign{
& \therefore \,\tan \left( { - \frac{\pi }{4}} \right) < \tan \,x < \tan \left( {\frac{\pi }{4}} \right) \cr
& \Rightarrow - 1 < \tan \,x < 1 \cr
& \Rightarrow 0 < {\tan ^2}x < 1 \cr
& \Rightarrow \left[ {{{\tan }^2}x} \right] = 0 \cr
& {\text{Hence, }}\mathop {\lim }\limits_{x \to 0} f\left( x \right) = \mathop {\lim }\limits_{x \to 0} \left[ {{{\tan }^2}x} \right] = 0 \cr
& {\text{Also }}f\left( 0 \right) = 0 \cr
& \therefore f\left( x \right)\,\,{\text{is continuous at}}\,x = 0 \cr} $$