To determine where $f(x)$ is continuous based on the graph, we analyze the following:

1. A function $f(x)$ is continuous at a point $x = c$ if:
   • $f(c)$ is defined.
   • The limit of $f(x)$ as $x \to c$ exists.
   • The value of the limit is equal to $f(c)$.

Observations:

1. Intervals of Continuity:

   • $f(x)$ appears to be continuous in all segments where the graph is unbroken (no gaps, jumps, or asymptotes).

2. Discontinuities:

   • At $x = -8$, there is a jump discontinuity as the left-hand and right-hand limits do not match.
   • At $x = 3$, there appears to be another discontinuity, possibly a removable one (hole in the graph).
   • At $x = 8$, there is another jump discontinuity.

Result:

The function $f(x)$ is continuous on intervals excluding these discontinuities: $\text{Continuity: } (-\infty, -8) \cup (-8, 3) \cup (3, 8) \cup (8, \infty)$

It is not continuous at $x = -8$, $x = 3$, and $x = 8$.

Would you like further clarification or analysis? Here are some related questions to expand on this topic:

1. What are the formal types of discontinuities (removable, jump, infinite)?
2. How can you determine continuity algebraically for a given function?
3. What role do limits play in analyzing continuity?
4. How can piecewise functions be analyzed for continuity?
5. What is the difference between continuity on an interval and at a point?

Tip: Always check both the left-hand and right-hand limits when evaluating continuity at specific points!