If f ( x ) = 2 x and g ( x ) = x 2 2 + 1 , then which of the following can be discontinuous functions?

Question

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

Smart Achievers · Accepted Answer

(4)

Ans. $\frac{g (x)}{f (x)}$

Explanation:

We know that, if $f$ and $g$ are continuous functions, then

(1) $f + g$ is continuous.

(2) $f - g$ is continuous.

(3) $f g$ is continuous.

(4) $\frac{f}{g}$ is continuous at those points where $g (x) \neq 0$

Here, $\frac{g (x)}{f (x)} = \frac{\frac{x^{2}}{2} + 1}{2 x} = \frac{x^{2} + 2}{4 x},$ which is discontinuous at $x = 0$ .

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