The correct answer is option 'D': Right continuous, left continuous, and continuous at each point of (a,b).

To understand why this is the correct answer, let's break down each condition and explain how it contributes to the overall continuity of a function on the interval (a,b).

1. The function is continuous at each point of (a,b):
This condition means that the function should not have any abrupt jumps or breaks at any point within the interval (a,b). In other words, the function should be defined and have a limit at every point in (a,b). This ensures that there are no sudden changes or discontinuities in the function's behavior, allowing for a smooth and continuous curve.

2. The function is right continuous:
Right continuity implies that the function should have a limit from the right at every point in the interval (a,b). In mathematical terms, for any point c in (a,b), the limit of the function as x approaches c from the right (x → c+) should exist and be equal to the value of the function at c. This condition ensures that there are no abrupt changes when approaching a point from the right side, contributing to the overall continuity of the function.

3. The function is left continuous:
Similarly, left continuity means that the function should have a limit from the left at every point in the interval (a,b). For any point c in (a,b), the limit of the function as x approaches c from the left (x → c-) should exist and be equal to the value of the function at c. This condition ensures that there are no abrupt changes when approaching a point from the left side, further contributing to the continuity of the function.

4. Continuous at each point of (a,b):
This condition combines the previous three conditions and requires the function to satisfy all of them. It ensures that the function is continuous at every point within the interval (a,b), both from the left and right sides. By meeting this condition, the function has a smooth and uninterrupted behavior throughout (a,b).

Overall, the correct answer is option 'D' because it includes all the necessary conditions for a function to be continuous on the interval (a,b). Each condition contributes to ensuring a smooth and uninterrupted curve, without any abrupt jumps or breaks, when considering the function's behavior at each point within the interval.