Understanding the Continuity Rule in Calculus: Definition and Application

Continuity Rule

The continuity rule is a fundamental concept in calculus that describes the behavior of a function and its graph

The continuity rule is a fundamental concept in calculus that describes the behavior of a function and its graph. It is used to determine whether a function is continuous or discontinuous at a given point.

To understand the continuity rule, let’s start with the definition of continuity. A function is said to be continuous at a point if three conditions are met:

1. The function is defined at that point. In other words, the value of the function is well-defined at the given point.

2. The limit of the function as it approaches that point exists. This means that as you get closer and closer to the given point, the function does not have any sudden jumps or gaps.

3. The value of the function at the given point is equal to the limit value. This ensures that there are no “holes” or “breaks” in the graph of the function at that point.

The continuity rule states that for any two functions f(x) and g(x) that are continuous at a given point x = a, the following operations will also result in a continuous function at point x = a:

1. Addition or subtraction: If f(x) and g(x) are continuous at x = a, then their sum or difference f(x) + g(x) or f(x) – g(x) will also be continuous at x = a.

2. Multiplication: If f(x) and g(x) are continuous at x = a, then their product f(x) * g(x) will also be continuous at x = a.

3. Division: If f(x) and g(x) are continuous at x = a, and g(a) ≠ 0, then their quotient f(x) / g(x) will also be continuous at x = a.

4. Composition: If f(x) and g(x) are continuous at x = a, then their composition f(g(x)) will also be continuous at x = a.

By applying the continuity rule, we can determine whether a given function will be continuous or discontinuous at a particular point. However, it is important to note that the continuity rule applies to functions that are already defined as continuous at a given point. It does not guarantee that a function will be continuous in general or at other points.

To determine the continuity of a function at a specific point, you may need to analyze its behavior algebraically or use various calculus techniques such as the limit definition or the intermediate value theorem.

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