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?

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.