1. **State the problem:** We need to determine if the piecewise function $$f(x) = \begin{cases} 2x - 1, & x < 5 \\ x^2, & x \geq 5 \end{cases}$$ is continuous on the interval $(5, 8]$. 2. **Recall the definition of continuity:** A function is continuous on an interval if it is continuous at every point in that interval. Since the interval is $(5, 8]$, we check continuity for all $x$ such that $5 < x \leq 8$. 3. **Check continuity for $x > 5$:** For $x > 5$, the function is defined as $f(x) = x^2$, which is a polynomial and continuous everywhere. Therefore, $f$ is continuous on $(5, 8]$. 4. **Check continuity at $x=5$:** Although $5$ is not included in the open interval $(5, 8]$, the question includes $5$ in the domain of the second piece ($x \geq 5$). Since the interval is $(5, 8]$, continuity at $5$ is not required for the interval but is relevant for the function's behavior. 5. **Conclusion:** On the interval $(5, 8]$, the function is continuous because for all $x > 5$, $f(x) = x^2$ is continuous, and the interval does not include $5$ itself (since it is open at 5). Thus, $f$ is continuous on $(5, 8]$. **Final answer:** The function $f$ is continuous on the interval $(5, 8]$.