Discuss the continuity of the function $f$ given by $f(x) = |x|$ at $x = 0$.

Continuous at $x = 0$ because $\lim\limits_{x \to 0} f(x) = f(0)$.

The correct answer is Option (2) → Continuous at $x = 0$ because $\lim\limits_{x \to 0} f(x) = f(0)$. ##

By definition

$f(x) = \begin{cases} -x, & \text{if } x < 0 \\ x, & \text{if } x \geq 0 \end{cases}$

Clearly the function is defined at 0 and $f(0) = 0$. Left hand limit of $f$ at 0 is

$\lim\limits_{x \to 0^-} f(x) = \lim\limits_{x \to 0^-} (-x) = 0$

Similarly, the right hand limit of $f$ at 0 is

$\lim\limits_{x \to 0^+} f(x) = \lim\limits_{x \to 0^+} x = 0$

Thus, the left hand limit, right hand limit and the value of the function coincide at $x = 0$. Hence, $f$ is continuous at $x = 0$.