Continuity Rule
A function is continuous at x = c if: (1) f(c) is defined (2) lim f(x) (x goes to c) exists (3) lim f(x) (x goes to c) = f(c)
The continuity rule is a mathematical concept used in calculus to determine whether a function is continuous. A function is said to be continuous when there are no sudden jumps or breaks in the graph. In other words, it means that the function can be drawn in one smooth line without any gaps, holes or jumps.
To check whether a function is continuous at a certain point, we need to make sure that three conditions are met:
1. The function is defined at the point.
2. The limit of the function as it approaches the point exists.
3. The limit of the function at the point is equal to the value of the function at the point.
If all three conditions are met, then the function is said to be continuous at that point. If any of the conditions are not met, then the function is said to be discontinuous at that point.
The continuity rule is important in calculus because it allows us to find the limits of functions and to determine whether a function is differentiable at a certain point. It also helps us to understand the behavior of functions over an interval.
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