Question

The function $f\left(x\right)=\left[x^2\right]+{[-x]}^2$, where $\left[\right]$ denotes the greatest integer function is

• A. Continuous and derivable at $x=2$
• B. Continuous nor derivable at $x=2$
• C. Continuous but not derivable at $x=2$
• D. None of the above

Answer 1

Answer: (B)

Continuous nor derivable at $x=2$

We have
$\underset{x\to 2^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(2-h)=\underset{h\to 0}{lim}{(2-h)}^2+{(-2+h)}^2$
$\Rightarrow \underset{x\to 2^{-}}{lim}f\left(x\right)=3+{(-2)}^2=7$
$\underset{x\to 2^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(2+h)=\underset{h\to 0}{lim}{(2+h)}^2+{(-2-h)}^2$
$\underset{x\to 2^{+}}{lim}f\left(x\right)=4+{(-3)}^2=13$
clearly, $\underset{x\to 2^{-}}{lim}f\left(x\right)\ne \underset{x\to 2^{+}}{lim}f\left(x\right)$
so, $f\left(x\right)$ is discontinuous at $x=2$
Consequently, it is not differentiable at $x=2$