Examine whether the function $f$ given by $f(x) = x^2$ is continuous at $x = 0$.

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

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

First note that the function is defined at the given point $x = 0$ and its value is 0. Then find the limit of the function at $x = 0$. Clearly

$\lim\limits_{x \to 0} f(x) = \lim\limits _{x \to 0} x^2 = 0^2 = 0$

Thus $\lim\limits _{x \to 0} f(x) = 0 = f(0)$

Hence, $f$ is continuous at $x = 0$.