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To determine the interval on which the function \( f \) is continuous, we need to examine the graph and identify where there are no breaks, jumps, or holes within the specific intervals provided.

### Analysis of the Options
1. **Option (A) (0, 1)**: In this interval, the function \( f \) has a jump discontinuity as it approaches \( x = 1 \) from the left and the right. The function does not meet in a continuous manner here.
  
2. **Option (B) (1, 2)**: There is an open circle at \( x = 1 \) and a filled circle at \( x = 2 \), indicating a discontinuity at \( x = 1 \). Therefore, the function is not continuous on this interval.

3. **Option (C) (2, 3)**: In this interval, the function appears smooth without any breaks, holes, or jumps. There is continuity across all points between \( x = 2 \) and \( x = 3 \).

4. **Option (D) (3, 4)**: There is a hole at \( x = 3 \) (an open circle), which makes the function discontinuous at that point.

### Conclusion
The only interval on which \( f \) is continuous is:

**Answer: (C) (2, 3)**

Would you like further explanation or have any questions?

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Here are some related questions to deepen understanding:

1. What is the definition of continuity at a point?
2. How can we determine continuity on a closed interval versus an open interval?
3. What types of discontinuities can exist in a function (e.g., jump, infinite, removable)?
4. How does one distinguish between open and closed circles in a graph?
5. Why is continuity important in the study of calculus?

**Tip:** When analyzing continuity in a graph, pay close attention to open and closed points, as they indicate whether the function actually "touches" that value at a given point.

The graph of the function f is shown above. On which of the following intervals is f continuous?
Options:
A. (0, 1)
B. (1, 2)
C. (2, 3)
D. (3, 4)

Learn how to determine intervals of continuity for a function based on its graph. This question explores continuity concepts on intervals with an example function, ideal for high school calculus students.

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