Understanding the Continuity Rule in Calculus: A Guide to Analyzing Function Continuity at Specific Points

Continuity Rule

The continuity rule, also known as the continuity theorem, is an important concept in calculus that helps us determine the continuity of a function at a particular point

The continuity rule, also known as the continuity theorem, is an important concept in calculus that helps us determine the continuity of a function at a particular point. It states that if a function is defined on a closed interval [a, b] and is continuous on the open interval (a, b), then the function is also continuous at the endpoints a and b, as long as the limit of the function at those points exists.

To understand this concept better, let’s break it down into three parts:

1. The function is defined on a closed interval [a, b]: This means that the function is defined and has values for all points between a and b, including the endpoints a and b.

2. The function is continuous on the open interval (a, b): This means that the function does not have any sudden jumps, holes, or vertical asymptotes in the interval (a, b). It is a smooth curve without any interruptions.

3. The limit of the function at the endpoints a and b exists: This means that the function approaches a specific value as it gets closer to the endpoints a and b. In other words, as x approaches a or b, the function does not have any abrupt changes or discontinuities.

By satisfying these three conditions, we can conclude that the function is continuous at the endpoints a and b. This is because the function behaves smoothly within the open interval (a, b), and the limit at the endpoints exists.

It is important to note that the continuity rule is a useful tool for analyzing the continuity of a function at specific points. However, it does not guarantee the continuity of the function at all points within the closed interval [a, b].

To determine the continuity of a function at a specific point, you can consider the following steps:

1. Check if the function is defined and has values at the given point.
2. Analyze the behavior of the function around the point to see if it has any discontinuities, jumps, or vertical asymptotes.
3. Determine if the limit of the function exists as x approaches the given point. If the limit exists and is equal to the value of the function at the point, then the function is continuous at that point. If the limit does not exist or is different from the value of the function at the point, then the function is not continuous at that point.

The continuity rule is a fundamental concept in calculus that helps us understand the behavior of functions and analyze their continuity. By applying this rule and considering the three conditions mentioned above, we can determine the continuity of a function at specific points within a closed interval.

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