conditions for continuity
For a function to be continuous at a point, three conditions must be met:
1. The function must be defined at the point in question (i.e., the value of x at the point must be in the domain of the function).
2. The limit as x approaches the point from both the left and right must exist.
3. The value of the limit as x approaches the point from both the left and right must be the same (i.e., the left and right limits must "meet" at the point).
If all three conditions are met, then the function is continuous at that point.
To determine the conditions for continuity of a function, we need to consider three main aspects:
1. The function must be defined at that point.
- The function must have a value assigned to it at the given point. In other words, the point must be in the domain of the function.
2. The limit must exist at that point.
- The limit of the function as it approaches the given point from both the left and right sides must exist. The value of the limit from both sides should also be equal. If the two-sided limit exists and is equal, we say that the limit at that point exists.
3. The value of the function must equal the limit at that point.
- The actual value of the function at the given point should be equal to the limit of the function at that point. If this condition is satisfied, we say that the function is continuous at that point.
Overall, for a function to be continuous at a particular point, the function must be defined at that point, the limit of the function at that point must exist, and the value of the function must equal the limit at that point.