To determine whether the function $f(x)$ is continuous on its domain, we need to examine both the graphical behavior and the formal definition of continuity. A function is continuous on its domain if, at every point within the domain, the limit of the function as $x$ approaches the point from both directions exists and is equal to the function's value at that point.

Let's break down the graph:

1. Point at $x = -1$:

   • The graph shows an open circle at $(-1, -1)$ and a closed point at $(-1, 1)$.
   • The left-hand limit as $x \to -1^-$ (from the left) is $\lim_{x \to -1^-} f(x) = -1$, as the graph approaches $(-1, -1)$.
   • The right-hand limit as $x \to -1^+$ (from the right) is $\lim_{x \to -1^+} f(x) = -1$.
   • However, $f(-1) = 1$, based on the closed point at $(-1, 1)$. Since $f(-1) \neq \lim_{x \to -1} f(x)$, the function is discontinuous at $x = -1$.

2. Point at $x = 0$:

   • The function has no value at $x = 0$ (there is no defined point there), but there are two different behaviors on either side:
     • As $x \to 0^+$ (from the right), the graph approaches a value of $2$.
     • As $x \to 0^-$ (from the left), the graph approaches a value near $-1$.
   • Since $\lim_{x \to 0^+} f(x) \neq \lim_{x \to 0^-} f(x)$, the limit does not exist at $x = 0$. This makes the function discontinuous at $x = 0$.

Conclusion:

The function $f(x)$ is not continuous on its domain. It is discontinuous at $x = -1$ because $\lim_{x \to -1} f(x) \neq f(-1)$, and at $x = 0$ because $\lim_{x \to 0} f(x)$ does not exist.

Would you like more details on any specific aspect? Here are some related questions:

1. What is the formal definition of continuity at a point?
2. How can you find limits graphically?
3. What are the types of discontinuities?
4. How would the function look if it were continuous?
5. What is the difference between removable and non-removable discontinuities?

Tip: When checking for continuity, always compare the left-hand limit, right-hand limit, and the function's value at the point!