Continuity Rule
The continuity rule is a concept in calculus that allows us to determine if a function is continuous at a specific point or within an interval
The continuity rule is a concept in calculus that allows us to determine if a function is continuous at a specific point or within an interval. Continuity refers to the absence of any “holes,” “jumps,” or “breaks” in the graph of a function.
To apply the continuity rule, we need to verify three conditions:
1. The function must be defined at the given point or within the interval we are interested in. In other words, there should be a valid value for the function at that point or within the interval.
2. The limit of the function as it approaches the given point or interval must exist. The limit is a value that the function gets arbitrarily close to as the independent variable gets arbitrarily close to the given point or interval.
3. The value of the function at the given point or within the interval must be equal to the limit value. This means that the value of the function “connects” smoothly to the limit value without any abrupt changes.
If all three conditions are met, we can conclude that the function is continuous at the given point or within the interval. This indicates that the graph of the function can be drawn without lifting the pen or the function can be completely traced without any interruptions.
On the other hand, if any of the conditions fail, we say that the function is not continuous at the given point or within the interval. This implies that there is some sort of discontinuity, such as a removable discontinuity, jump discontinuity, or an infinite discontinuity.
The continuity rule is a fundamental concept in calculus as it helps us understand the behavior of functions and their graphs. By identifying where a function is continuous or discontinuous, we can make predictions and analyze various properties of functions in different mathematical and real-life contexts.
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