This problem asks for a function $f$ that satisfies various conditions based on limits and function values, and the sketch of the function is shown on the graph.

Here’s a breakdown of what the question is asking:

Conditions and Explanation:

1. $f(1) = 1$ and $f(3) = 1$:

   • These conditions are marked as correct (✓✓). On the graph, $f(1)$ and $f(3)$ both equal 1, which matches the requirement.

2. $\lim_{x \to 1^-} f(x) = 1$ and $\lim_{x \to 1^+} f(x) = 0$:

   • This is marked as incorrect (X). According to the sketch, the left-hand limit as $x \to 1^-$ is 1, but the right-hand limit $x \to 1^+$ is not 0 (it’s closer to 1), contradicting the problem requirement.

3. $\lim_{x \to 2^-} f(x) = \infty$ and $\lim_{x \to 2^+} f(x) = \infty$:

   • Marked as correct (✓✓). The graph shows a vertical asymptote at $x = 2$, where the function heads to infinity from both sides, matching the condition.

4. $\lim_{x \to 3^-} f(x) = 1$ and $\lim_{x \to 3^+} f(x) = 1$:

   • This is half correct (✓X). The limit from the left-hand side is 1 (correct), but from the right-hand side, the graph shows a discrepancy, as the function doesn't exactly match 1.

5. Continuity:

   • $f$ should be continuous except at $x = 1$, $x = 2$, and $x = 3$. The function is correct in terms of the graph not being continuous at $x = 2$ (due to the vertical asymptote) and at $x = 1$ (due to the different left and right limits). The teacher has crossed out continuity at $x = 3$, possibly due to the right-hand limit issue.

Graph Analysis:

The graph depicts:

• A hole at $x = 1$, implying the function is not continuous there, and the left and right-hand limits are not both equal to 1.
• A vertical asymptote at $x = 2$, where the function approaches infinity.
• Behavior near $x = 3$ is almost continuous but doesn’t satisfy the condition fully, which explains the teacher’s "XX" mark.

Would you like more details on specific steps, or further analysis of any points?

Here are 5 questions to consider:

1. What is the significance of left-hand and right-hand limits in determining continuity?
2. How do vertical asymptotes impact the continuity of a function?
3. Why is it necessary for the limits from both sides at $x = 3$ to match the function value?
4. How can a function be discontinuous at a point yet still be defined at that point?
5. What techniques can be used to analyze limits graphically?

Tip: Always verify both the left and right limits when checking for continuity or matching conditions at specific points!