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 $x = a$.

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.