I see you have uploaded an image containing a function with some questions about continuity. Let me break down the information for you:

Part (a): $x = 2$

The function at $x = 2$ has a break, so the condition that fails is:

• Condition 2: $f(2)$ does not exist.

This is a discontinuity where the function is not defined at $x = 2$.

Part (b): $x = 5$

At $x = 5$, the function has a hole, meaning:

• Condition 3: $\lim_{x \to 5} f(x) \neq f(5)$ fails. The limit exists but does not match the value of the function at that point.

Part (c): $x = 9$

At $x = 9$, the function is continuous:

• $f$ is continuous.

The limit at $x = 9$ exists, the function value is defined at $x = 9$, and the limit matches the function value.

Would you like further clarification on any of these results, or do you have other questions? Here are some follow-up questions:

1. What does it mean for a function to be continuous at a point?
2. How do you visually identify discontinuities from a graph?
3. What are the three conditions of continuity?
4. How can we fix discontinuities in a function?
5. What are some real-world examples of functions that might have discontinuities?

Tip: To check for continuity, always verify if the function is defined at a point and whether the limit matches the function value at that point.