Question

The function $f\left(x\right)=\frac{4-x^2}{4x-x^3}$
(a) discontinuous at only one point
(b) discontinuous exactly at two points
(c) discontinuous exactly at three points
(d) none of these

Answer 1

(C) discontinuous exactly at three points

Given: $f\left(x\right)=\frac{4-x^2}{4x-x^3}$
$\Rightarrow f\left(x\right)=\frac{4-x^2}{x\left(4-x^2\right)}$
$\Rightarrow f\left(x\right)=\frac{1}{x},x\ne 0\mathrm{and}4-x^2\ne 0\mathrm{or}x\ne 0,\pm 2$

Clearly, $f\left(x\right)$ is defined and continuous for all real numbers except $\left\{0,\pm 2\right\}$.

Therefore, $f\left(x\right)$ is discontinuous exactly at three points.