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) → $\frac{g(x)}{f(x)}$

f is not discontinuous g is not discontinuous

$⇒fg,f+g,f-g$ can never be discontinuous

but g/f is discontinuous 

at $f=0$, at $x=0$

since g/f is not defined at x = 0