If f (x) = 2x and g (x) = x²/2 + 1, then which of the following can be a discontinuous function

The correct option is g(x)/f(x).

Step-by-step Explanation:

We are given the following two functions:
$$f(x)=2x$$
and
$$g(x)=\frac{x^2}{2}+1$$

Both $f(x)$ and $g(x)$ are polynomial functions. A fundamental property of polynomial functions is that they are continuous everywhere on the set of real numbers ($ℝ$).

According to the algebra of continuous functions, if two functions $f(x)$ and $g(x)$ are continuous on a domain, then:
1. Their sum, $f(x)+g(x)$, is continuous.
2. Their difference, $f(x)-g(x)$, is continuous.
3. Their product, $f(x)\cdot g(x)$, is continuous.
4. Their quotient, $\frac{g(x)}{f(x)}$, is continuous except at points where the denominator becomes zero ($f(x)=0$).

Let's examine the quotient function:
$$h(x)=\frac{g(x)}{f(x)}=\frac{\frac{x^2}{2}+1}{2x}$$

For this quotient function to be defined, the denominator must not be zero. We find the point of discontinuity by setting the denominator to zero:

$$2x=0\Rightarrow x=0$$

Since the function $\frac{g(x)}{f(x)}$ is undefined at $x=0$, it is discontinuous at that point. Thus, g(x)/f(x) can be a discontinuous function.