key term - |x - a| < δ

Definition

|x - a| < δ is a mathematical expression that defines the concept of continuity. It represents the idea that a function is continuous at a point 'a' if the distance between the function's value at 'x' and the function's value at 'a' can be made arbitrarily small by choosing 'x' sufficiently close to 'a'. This is a crucial concept in the study of calculus and the analysis of the behavior of functions.

5 Must Know Facts For Your Next Test

1. The expression |x - a| < δ is used to define the continuity of a function at a specific point 'a'.
2. The value of 'δ' represents the maximum allowable distance between the function's value at 'x' and the function's value at 'a' for the function to be considered continuous at 'a'.
3. If a function satisfies the condition |x - a| < δ, it means the function's value at 'x' can be made arbitrarily close to the function's value at 'a' by choosing 'x' sufficiently close to 'a'.
4. Continuity is an important property of functions, as it ensures the function's values change smoothly and predictably as the input variable changes.
5. The epsilon-delta definition of continuity, which uses the expression |x - a| < δ, provides a rigorous mathematical framework for analyzing the continuity of functions.

Review Questions

• Explain the significance of the expression |x - a| < δ in the context of function continuity.
  • The expression |x - a| < δ is a key part of the formal definition of continuity for a function at a specific point 'a'. It states that the function is continuous at 'a' if the absolute difference between the function's value at 'x' and the function's value at 'a' can be made arbitrarily small by choosing 'x' sufficiently close to 'a'. This ensures that the function's values change smoothly and predictably as the input variable changes, which is a crucial property for many applications of calculus and mathematical analysis.
• Describe how the value of 'δ' relates to the continuity of a function at a point 'a'.
  • The value of 'δ' in the expression |x - a| < δ represents the maximum allowable distance between the function's value at 'x' and the function's value at 'a' for the function to be considered continuous at 'a'. A smaller value of 'δ' means the function's values at 'x' must be closer to the function's value at 'a' for the function to be continuous. The ability to choose a sufficiently small 'δ' value is what defines the continuity of a function at a point 'a' in the epsilon-delta definition of continuity.
• Analyze how the expression |x - a| < δ relates to the concept of limits and the behavior of a function as 'x' approaches 'a'.
  • The expression |x - a| < δ is closely tied to the concept of limits in calculus. As 'x' approaches the point 'a', the absolute difference |x - a| must become arbitrarily small, which is captured by the condition |x - a| < δ. This ensures that the function's value at 'x' can be made arbitrarily close to the function's value at 'a' as 'x' gets closer to 'a'. This relationship between the function's values and the distance between 'x' and 'a' is the essence of the epsilon-delta definition of continuity, which provides a rigorous mathematical framework for analyzing the behavior of functions as input values approach a particular point.

Related terms

Limit:

The limit of a function as the input approaches a particular value is the value that the function approaches as the input gets closer and closer to that value.

Epsilon-Delta Definition of Continuity: The formal definition of continuity at a point 'a' using the expression |x - a| < δ, where 'δ' is a positive real number that can be made arbitrarily small.

Function Continuity: A function is continuous at a point if it satisfies the condition |x - a| < δ, meaning the function's value at 'x' can be made arbitrarily close to the function's value at 'a' by choosing 'x' sufficiently close to 'a'.