Discuss the continuity of the function f defined by f(x) = 1/x, x ≠ 0.

The correct answer is Option (2) → Continuous at every point in its domain $(x \neq 0)$. ##

Fix any non zero real number $c$, we have

$\lim\limits_{x \to c} f(x) = \lim\limits_{x \to c} \frac{1}{x} = \frac{1}{c}$

Also, since for $c \neq 0$, $f(c) = \frac{1}{c}$, we have $\lim\limits_{x \to c} f(x) = f(c)$ and hence, $f$ is continuous at every point in the domain of $f$. Thus $f$ is a continuous function.