Step-by-step Solution:

Given that: \[ f(x) = \begin{cases} 0, & \text{if } x \leq 0, \\ 2x, & \text{if } x > 0, \end{cases} \] Now, we check the continuity of the given function at \( x = 0 \). The left-hand limit (LHL) is: \[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0} 0 = 0. \] The right-hand limit (RHL) is: \[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0} 2x = 2 \times 0 = 0. \] And since \( f(0) = 0 \), we have: \[ \text{LHL} = \text{RHL} = f(0) = 0. \] Hence, the given function is continuous at \( x = 0 \). Therefore, the given function is continuous in the interval \( (-\infty, \infty) \).