If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$, then which of the following can be a discontinuous function?

$\frac{g(x)}{f(x)}$

The correct answer is Option (4) → $\frac{g(x)}{f(x)}$ ##

We know that, if $f$ and $g$ be continuous functions, then

(a) $f + g$ is continuous

(b) $f - g$ is continuous

(c) $fg$ is continuous

(d) $\frac{f}{g}$ is continuous at those points, where $g(x) \neq 0$.

Here,

$\frac{g(x)}{f(x)} = \frac{\frac{x^2}{2} + 1}{2x} = \frac{x^2 + 2}{4x}$

which is discontinuous at $x = 0$.