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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.

Analyze the function f(x) at x = 2 and determine whether f(2) is defined, whether the limit exists, and if the function is continuous at x = 2.

Explore how to analyze the continuity of a function at x = 2 by determining whether f(2) is defined, if the limit exists, and whether the function is continuous. This problem covers essential calculus concepts, including limits and the definition of continuity.

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