https://file.aiquickdraw.com/mathbot/file/173894827969414e0g2ee.jpg

https://file.aiquickdraw.com/mathbot/file/1738948279695gz8o4ppw.jpg

From the graphs provided and the information in the first image, you're being asked to evaluate the continuity of a function \( f(x) \) at \( x = 0 \).

### Observations:
1. The first image contains options to check the behavior of the function at \( x = 0 \).
2. In the second image, the graph of \( y = f(x) \) is shown with a visible discontinuity at \( x = 0 \). Specifically, there is a hole in the graph at \( x = 0 \), indicating that the function is **not defined** at that point.

### Analysis of the options:
- **f is continuous**: This is **incorrect** because there is a discontinuity at \( x = 0 \), so the function is not continuous.
- **lim \( f(x) \) does not exist as \( x \to 0 \)**: This is **incorrect** because the limit appears to exist from both sides, but the function does not match the limit exactly at \( x = 0 \).
- **f(0) does not exist**: This is **correct** because the function has a hole at \( x = 0 \), meaning that \( f(0) \) is not defined.
- **lim \( f(x) \neq f(0) \)**: This is **incorrect** because the limit from both sides at \( x = 0 \) exists, but since \( f(0) \) is undefined, the limit does not match any specific function value.

### Correct Answer: **f(0) does not exist**

Would you like further clarification on any specific part of this process?

### Additional Related Questions:
1. What is the definition of continuity at a point?
2. How do you evaluate the limit of a function at a specific point?
3. What are the different types of discontinuities in functions?
4. How can we formally define a hole in a function on a graph?
5. What is the importance of one-sided limits in analyzing discontinuities?

**Tip:** When analyzing continuity, always check if the function is defined at the point of interest and if the limit exists and matches the function value at that point.

Consider the following graph and determine the correct statement about the function at x = 0.

Learn how to determine the continuity of a function at x = 0 based on its graph. This problem involves analyzing the existence of limits and function values to assess the behavior of a function at a specific point. Suitable for students in Grades 11-12.

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