Question

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

• A. $f\left(x\right)+g\left(x\right)$
• B. $f\left(x\right)–g\left(x\right)$
• C. $f\left(x\right).g\left(x\right)$
• D. $\frac{g\left(x\right)}{f\left(x\right)}$
• E. $\frac{f\left(x\right)}{g\left(x\right)}$

Answer 1

The correct option is D $\frac{g\left(x\right)}{f\left(x\right)}$

The functions $f\left(x\right)=2x$ and $g\left(x\right)=\frac{x^2}{2}+1$ are polynomial functions and hence, are continuous everywhere.
Therefore,$f\left(x\right)+g\left(x\right),f\left(x\right)-g\left(x\right),f\left(x\right).g\left(x\right)$ are continuous everywhere.
$\frac{g\left(x\right)}{f\left(x\right)}$ is discontinuous where $f\left(x\right)=0$, i.e. x = 0.
$\frac{f\left(x\right)}{g\left(x\right)}$ is discontinuous where $g\left(x\right)=0$, i.e. $\frac{x^2}{2}+1=0\Rightarrow x^2=-2$
No such value of x exists. So, $\frac{f\left(x\right)}{g\left(x\right)}$ is continuous everywhere.