Analyzing Function Continuity: Applying The Fundamental Continuity Rule In Calculus

Continuity Rule

If the limit exists (aka left limit and right limit are equal), and the limit equals the function at that point.

The continuity rule is a concept in calculus that provides a criterion for determining whether a function is continuous at a specific point in its domain. In essence, the continuity rule states that for a function f(x) to be continuous at a point c in its domain, three conditions must be met:

1. f(c) must be defined
2. The limit of f(x) as x approaches c must exist,
3. The limit of f(x) as x approaches c must be equal to f(c).

If a function fails to meet any of these conditions, then it is not continuous at that specific point.

The continuity rule is an essential concept in calculus because it helps us to understand the behavior of functions and their graphs. For example, a function that is continuous at every point in its domain will have a smooth and well-behaved graph without any sudden jumps or discontinuities. In contrast, a function that is not continuous at a point may have a graph with sharp corners or sudden jumps.

To apply the continuity rule, we can use algebraic manipulations and the properties of limits to evaluate the limit of the function at the point of interest. If the limits from both the left and right sides of the point are equal and finite, then the function is continuous there. If either one side limit does not exist or the function value is different, the function is not continuous at that point.

In summary, the continuity rule is a fundamental concept in calculus that provides a criterion for determining whether a function is continuous at a specific point in its domain. The rule helps us to understand the behavior of functions and their graphs and is essential for solving a wide range of problems in applied mathematics.

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