CUET Preparation Today
Discuss the continuity of the function $f$ defined by $f(x) = \frac{1}{x}, x \neq 0$. |
Continuous for all $x \in \mathbb{R}$. Continuous at every point in its domain $(x \neq 0)$. Discontinuous everywhere. Continuous only for positive values of $x$. |
Continuous at every point in its domain $(x \neq 0)$. |
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. |
