Answer
Please see below.
Work Step by Step
The graph does not appear to be continuous on the interval $[-4,4]$ because it has a hole at $(2,12)$.
We can argue analytically that the function $f(x)= \frac{x^3-8 }{x-2}$ is not continuous on the interval $[-4,4]$ because the function is not defined at $x=2$.
Graphs are good tools for examining continuity of functions. When the graph of a function does not have any hole and jump on an interval, we can conclude that the function is continuous on the interval. However, we should note that we cannot always judge continuity of functions from their graphs; for example, the graph of some functions have infinitely many fluctuations on small intervals (for example, $g(x)= \sin \frac{1}{x}$). So, in such cases, analytical methods are required. Please also note that deduction by graphs is not considered as a mathematical proof.
