The function $f(x)=\left[x^2\right]+[-x]^2$, where [.] denotes the greatest integer function, is
Answer & explanation
Correct answer: option 2
We have,
$\lim\limits_{x \rightarrow 2^{-}} f(x)=\lim\limits_{h \rightarrow 0} f(2-h)=\lim\limits_{h \rightarrow 0}\left[(2-h)^2\right]+[-2+h]^2$
$\Rightarrow \lim\limits_{x \rightarrow 2^{-}} f(x)=3+(-2)^2=7$
$\lim\limits_{x \rightarrow 2^{+}} f(x)=\lim\limits_{h \rightarrow 0} f(2+h)=\lim\limits_{h \rightarrow 0}\left[(2+h)^2\right]+[-2-h]^2$
$\Rightarrow \lim\limits_{x \rightarrow 2^{+}} f(x)=4+(-3)^2=13$
Clearly, $\lim\limits_{x \rightarrow 2^{-}} f(x) \neq \lim\limits_{x \rightarrow 2^{+}} f(x)$
So, f(x) is discontinuous at x = 2.
Consequently, it is non-differentiable at x = 2.