To determine the intervals of continuity for the given function based on the graph description, we need to analyze the behavior of the function in different regions.

Key observations from the graph description:

1. First branch (left of $x = 1$):

   • Starts at an open circle at $(0, 1)$, indicating a discontinuity at $x = 0$.
   • The function rises steeply toward infinity as $x$ approaches 1 from the left. This suggests an infinite discontinuity at $x = 1$ from the left.

2. Second branch (between $x = 1$ and $x = 2$):

   • The function is defined for $1 < x < 2$, with a vertical asymptote at $x = 1$.
   • It ends at an open circle at $(2, 1)$, indicating a discontinuity at $x = 2$.

3. Third branch (between $x = 2$ and $x = 5$):

   • Starts at a closed circle at $(2, 2)$, suggesting continuity from the left at $x = 2$.
   • The function is continuous from $x = 2$ to $x = 5$, including the points $(3, 3)$, $(4, 4)$, and $(5, 1)$, but ends with an open circle at $(5, 1)$, indicating a discontinuity at $x = 5$.

Conclusion:

The function is continuous on the following intervals:

• $(0, 1)$ (continuous on the left of 1, but there's a discontinuity at $x = 1$)
• $(2, 5)$ (continuous from 2 to 5, but with a discontinuity at $x = 5$)

Final answer in interval notation:

The intervals of continuity are: $(0, 1), (2, 5)$

For the endpoints of these intervals:

• At $x = 1$, the function is continuous from the right (approaching from the second branch).
• At $x = 2$, the function is continuous from the left (approaching from the third branch).
• At $x = 5$, the function is continuous from the left (approaching from the third branch).

Would you like more details on the behavior of the function at these specific points, or do you have any other questions?

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Here are some related questions you might find useful:

1. How do infinite discontinuities affect the continuity of a function at a given point?
2. What are open and closed circles on a graph, and how do they relate to continuity?
3. How do we define a function's continuity at a point in terms of limits?
4. What is the difference between a function being continuous from the left versus from the right?
5. How can we use the Intermediate Value Theorem to check for continuity on an interval?

Tip: When analyzing a graph for continuity, pay special attention to open and closed circles, vertical asymptotes, and behavior at the boundaries of intervals.