To determine the intervals of continuity for the function $f(x)$, we need to carefully examine the graph's behavior and identify where the function is continuous. Continuity means there are no breaks, jumps, or holes in the graph at a given point.

Key Observations:

1. From $x = 0$ to $x = 1$:

   • The function starts at an open circle at $(0, 4)$, which suggests $f(x)$ is not defined at $x = 0$.
   • It rises to another open circle at $(1, 5)$, suggesting the function is continuous on this segment but does not include $x = 0$ or $x = 1$.

2. From $x = 1$ to $x = 2$:

   • The graph falls from $(1, 5)$ to an open circle at $(2, 4)$.
   • This means the function is continuous between $x = 1$ and $x = 2$, but it has a jump discontinuity at $x = 2$, where there is an open circle at $(2, 4)$ and a closed circle at $(2, 5)$.

3. From $x = 2$ to $x = 3$:

   • The graph goes from the closed circle at $(2, 5)$ to the closed circle at $(3, 1)$.
   • This segment is continuous, as the function is defined and continuous for all points between $x = 2$ and $x = 3$.

4. From $x = 3$ to $x = 4$:

   • The function starts at an open circle at $(3, 4)$, which suggests a discontinuity at $x = 3$.
   • It rises steeply to the open circle at $(4, 5)$, so there is a jump discontinuity at $x = 3$.

5. From $x = 4$ to $x = 6$:

   • The graph is continuous between $x = 4$ and $x = 6$, as there are no further discontinuities.

Conclusion:

The function is continuous on the following intervals:

• $(1, 2)$
• $(2, 3)$
• $(4, 6)$

Continuity at Endpoints:

• At $x = 1$: The function is continuous from the right (i.e., it approaches the value of 5 as $x$ approaches 1 from the right).
• At $x = 2$: The function is continuous from the left (i.e., it approaches the value of 5 as $x$ approaches 2 from the left).
• At $x = 3$: The function is continuous from the left (i.e., it approaches the value of 4 as $x$ approaches 3 from the left).
• At $x = 4$: The function is continuous from the left (i.e., it approaches the value of 5 as $x$ approaches 4 from the left).

Final Answer:

The intervals of continuity for the given function are: $(1, 2), (2, 3), (4, 6)$

Let me know if you'd like any further details or clarifications!

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Here are some related questions you might find interesting:

1. How can you determine if a function is continuous at a specific point from its graph?
2. How do open and closed circles on a graph affect the continuity of a function?
3. What is the difference between continuity from the left and from the right?
4. What types of discontinuities might appear in piecewise functions?
5. How does the concept of a jump discontinuity differ from an infinite discontinuity?

Tip: Always check for breaks, jumps, or holes in the graph to identify points of discontinuity.