If $f(x)=2 x$ and $g(x)=\frac{x^{2}}{2}+1$, then which of the following can be a discontinuous function
(a) $f(x)+g(x)$
(b) $f(x)-g(x)$
(c) $f(x) g(x)$
(d) $\frac{g(x)}{f(x)}$
$f(x)=2 x$ and $g(x)=\frac{x^{2}}{2}+1$ are polynomial functions. We know polynomial functions are continuous for all values of $x$.
So, $f(x)=2 x$ and $g(x)=\frac{x^{2}}{2}+1$ are continuous functions.
Also, sum, difference and product of continuous functions is continuous functions.
$\therefore f(x)+g(x), f(x)-g(x)$ and $f(x) g(x)$ are continuous functions.
Now, $\frac{g(x)}{f(x)}$ is continuous if $f(x) \neq 0$.
But, $f(x)=2 x=0$ when $x=0 .$ So, $\frac{g(x)}{f(x)}$ is discontinuous at $x=0$
Thus, $\frac{g(x)}{f(x)}$ can be a discontinuous function.
Hence, the correct answer is option (d).
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