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.