To determine where the function $f(x)$ is continuous based on the graph you provided, we need to identify the points or intervals where the function is not continuous. Typically, discontinuities occur in the form of jumps, holes, or vertical asymptotes.

Here are the observations from the graph:

1. There is a vertical asymptote near $x = 1$, indicating that the function is not continuous at $x = 1$.
2. There is a hole (removable discontinuity) at $x = 3$, suggesting that the function is not continuous at this point either.

Aside from these discontinuities, the graph appears continuous elsewhere.

Interval Notation for Continuity

Thus, the function is continuous in the following intervals: $(-\infty, 1) \cup (1, 3) \cup (3, 5]$ This means that $f(x)$ is continuous everywhere except at $x = 1$ (asymptote) and $x = 3$ (hole).

Would you like further details on the types of discontinuities, or do you have any questions about the function's behavior?

Here are five related questions you might consider:

1. What types of discontinuities are there in a function?
2. How do you determine the limits near discontinuities?
3. What does it mean for a function to be continuous at a point?
4. How do we remove a discontinuity from a function like the one at $x = 3$?
5. How do asymptotes affect the continuity of a function?

Tip: When examining a graph, always check for vertical asymptotes, jumps, or holes to identify discontinuities.