Limit Definition of Continuity

Definition

The limit definition of continuity describes a function as continuous at a point if the function's limit at that point equals the function's value at that point. This means that as the input values approach a specific point, the function's output values also approach the function's value at that point, without any sudden jumps or breaks in the function's graph.

5 Must Know Facts For Your Next Test

1. For a function to be continuous at a point, the function's limit at that point must equal the function's value at that point.
2. Discontinuities occur when the function's limit at a point does not equal the function's value at that point.
3. One-sided limits, from the left or right, can be used to determine if a function is continuous at a point.
4. Continuous functions have no sudden jumps or breaks in their graphs, and their output values approach the function's value at a point as the input values approach that point.
5. The limit definition of continuity is a fundamental concept in calculus and is used to analyze the behavior of functions at specific points.

Review Questions

• Explain the relationship between the limit of a function and the function's value at a point in the context of continuity.
  • For a function to be continuous at a point, the function's limit at that point must equal the function's value at that point. This means that as the input values approach the point, the function's output values also approach the function's value at that point, without any sudden jumps or breaks in the function's graph. If the limit and the function's value at a point are not equal, then the function is discontinuous at that point.
• Describe how one-sided limits can be used to determine the continuity of a function at a point.
  • One-sided limits, from the left or right, can be used to determine if a function is continuous at a point. If the left-hand limit and the right-hand limit both exist and are equal to the function's value at that point, then the function is continuous at that point. However, if the left-hand limit and the right-hand limit are not equal, or if one or both of the one-sided limits do not exist, then the function is discontinuous at that point.
• Analyze the importance of the limit definition of continuity in the study of calculus and the behavior of functions.
  • The limit definition of continuity is a fundamental concept in calculus because it allows for the analysis of the behavior of functions at specific points. Continuous functions have no sudden jumps or breaks in their graphs, and their output values approach the function's value at a point as the input values approach that point. This continuity is essential for many calculus concepts, such as differentiation and integration, which rely on the smooth and predictable behavior of functions. Understanding the limit definition of continuity is crucial for studying the properties and characteristics of functions in calculus.

Related terms

Limit:

The value that a function's output approaches as the input values get closer and closer to a particular point.

Discontinuity:

A point where a function is not continuous, meaning the function's limit at that point does not equal the function's value at that point.

One-Sided Limits: The limits of a function approached from the left or right side of a point, which may or may not be equal to the function's value at that point.