https://file.aiquickdraw.com/mathbot/file/1726210590657aae7aohi.jpg

Looking at the graph provided and the given options, let's analyze the statements one by one:

1. **Statement A: "f is continuous at \( x = a \)"**
   - From the graph, we can see that there is a discontinuity at \( x = a \) because there is a hole (open circle) at that point. The function does not take a value at \( x = a \), so **this statement is false**.

2. **Statement B: "f has a relative maximum at \( x = a \)"**
   - The function seems to reach a peak at \( x = a \), indicating a relative maximum. Thus, **this statement is true**.

3. **Statement C: "x = a is in the domain of f"**
   - Since the graph has a hole at \( x = a \), it means the function is not defined at this point. So, **this statement is false**.

4. **Statement D: "The right-hand limit and left-hand limit of \( f \) as \( x \) approaches \( a \) are equal"**
   - Both the left-hand and right-hand limits appear to approach the same value as \( x \) approaches \( a \), even though the function is not defined at \( x = a \). Hence, **this statement is true**.

5. **Statement E: "The limit of \( f(x) \) as \( x \) approaches \( a \) exists"**
   - Since both the left-hand and right-hand limits exist and are equal, the limit exists. Thus, **this statement is true**.

### Conclusion:
The false statement is **A**. The function is not continuous at \( x = a \).

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Do you want more details or have any further questions? Here are some related questions:

1. What is the formal definition of continuity at a point?
2. How do we calculate left-hand and right-hand limits?
3. What are the necessary conditions for a relative maximum or minimum?
4. Why is it important that limits from both sides agree for the limit to exist?
5. How do discontinuities affect the behavior of a function?

**Tip:** Always check for open and closed circles in graphs—they indicate whether a function is defined at that point.

The graph of a function f is shown above. Which of the following statements about f is false?

This problem analyzes the function's continuity, limit behavior, and relative extrema based on a graph. Key concepts include limits, the definition of continuity, and relative maximum. Suitable for students in advanced high school calculus or pre-calculus courses.

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