Let's analyze the graph of the function $f(x)$ at $x = 2$ and address each of the given statements.

1. $f(2)$ is defined.

   • Looking at the graph, at $x = 2$, there is a hollow circle (indicating the function is not defined there), but there is a solid dot at $x = 2$ corresponding to $f(2) = 1$. Therefore, $f(2)$ is defined as 1.

2. $\lim_{x \to 2} f(x)$ exists.

   • To check if the limit exists at $x = 2$, we need to check the left-hand and right-hand limits.
   • As $x \to 2^-$ (from the left), $f(x)$ approaches 0.
   • As $x \to 2^+$ (from the right), $f(x)$ also approaches 0.
   • Since both limits are the same, the limit exists and equals 0.

3. $\lim_{x \to 2} f(x) = f(2)$.

   • From the previous points, we know $\lim_{x \to 2} f(x) = 0$ and $f(2) = 1$. Since these two values are not equal, this statement is false.

4. The function is continuous at $x = 2$.

   • For a function to be continuous at $x = 2$, three conditions must hold:
     1. $f(2)$ is defined (this is true; $f(2) = 1$).
     2. $\lim_{x \to 2} f(x)$ exists (this is true; the limit is 0).
     3. $\lim_{x \to 2} f(x) = f(2)$ (this is false because the limit is 0 and $f(2) = 1$).
   • Therefore, the function is not continuous at $x = 2$.

5. The function is not continuous at $x = 2$.

   • Since the function fails the condition $\lim_{x \to 2} f(x) = f(2)$, the function is indeed not continuous at $x = 2$.

Conclusion:

The correct statements are:

• $f(2)$ is defined.
• $\lim_{x \to 2} f(x)$ exists.
• The function is not continuous at $x = 2$.

Would you like any more details on these answers?

Here are 5 related questions for further practice:

1. What are the conditions for a function to be continuous at a point?
2. What is the difference between one-sided and two-sided limits?
3. How can you tell if a function is discontinuous at a point?
4. Can a function have a limit at a point but not be continuous at that point?
5. What happens to the graph of a function if the limit at a point does not exist?

Tip: Always check the one-sided limits when determining whether a limit exists at a point.