Step-by-step Solution:

We are given the function: \[ f(x) = \begin{cases} \frac{\sin x}{x}, & x \neq 0, \\ 0, & x = 0. \end{cases} \] Continuity of \(f(x)\) at \(x = 0\) To check continuity at \(x = 0\), we evaluate the left-hand limit, right-hand limit, and the function value \(f(0)\). \[ \lim_{x \to 0} f(x) = \lim_{x \to 0} \frac{\sin x}{x}. \] Using the standard limit result: \[ \lim_{x \to 0} \frac{\sin x}{x} = 1. \] The value of \(f(0)\) is given as 0. Since: \[ \lim_{x \to 0} f(x) \neq f(0), \] \(f(x)\) is not continuous at \(x = 0\).