Question

If f(x) = 2x and $g\left(x\right)=\frac{x^2}{2}+1,$ 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) $\frac{g\left(x\right)}{f\left(x\right)}$

Answer 1

f(x) = 2x 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) = 2x and g(x) = $\frac{x^2}{2}+1$ are continuous functions.

Also, sum, difference and product of continuous functions is continuous functions.

∴ f(x) + g(x), f(x) − g(x) and f(x)g(x) are continuous functions.

Now, $\frac{g\left(x\right)}{f\left(x\right)}$ is continuous if f(x) ≠ 0.

But, f(x) = 2x = 0 when x = 0. So, $\frac{g\left(x\right)}{f\left(x\right)}$ is discontinuous at x = 0.

Thus, $\frac{g\left(x\right)}{f\left(x\right)}$ can be a discontinuous function.

Hence, the correct answer is option (d).