From the graph provided, we need to determine where the function $f(x)$ is continuous. Here's the process:

1. Definition of Continuity: A function is continuous at a point $c$ if:

   • $f(c)$ is defined.
   • The limit of $f(x)$ as $x$ approaches $c$ exists.
   • The limit of $f(x)$ as $x$ approaches $c$ equals $f(c)$.

2. Analyzing the Graph:

   • Identify breaks, jumps, or holes in the graph.
   • Look for points where $f(x)$ is not defined or where there are abrupt discontinuities.

3. Observations from the Graph:

   • There appears to be a discontinuity at $x = -4$ (a vertical asymptote or undefined point).
   • There is a jump discontinuity at $x = 2$, as the function jumps from one value to another abruptly.
   • At $x = 6$, there seems to be a hole in the graph (open circle), indicating the function is not defined at that point.

4. Intervals of Continuity: Based on these observations, $f(x)$ is continuous on the following intervals:

   • $(-\infty, -4)$
   • $(-4, 2)$
   • $(2, 6)$
   • $(6, \infty)$

   The function is discontinuous at $x = -4$, $x = 2$, and $x = 6$.

If you have further questions or want more details, feel free to ask! Here are 5 related questions to expand on this:

1. How do we mathematically define limits at points of discontinuity?
2. What types of discontinuities exist, and how are they identified on a graph?
3. How can we test for removable vs. non-removable discontinuities?
4. What are some real-world examples of functions with discontinuities?
5. Can discontinuous functions still have derivatives at certain points?

Tip: To confirm continuity, always check both the limit from the left and the right at critical points!