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 mathematics that helps us determine the continuity of a function at a specific point or over a given interval. It is often used in calculus and analysis to study the behavior of functions and their limits.
Formally, a function f(x) is said to be continuous at a point x = c if three conditions are satisfied:
1. The function is defined at the point c.
2. The limit of the function exists as x approaches c.
3. The value of the function at c is equal to the limit.
Mathematically, we can express this as:
1. f(c) is defined and exists.
2. lim(x→c) f(x) exists.
3. lim(x→c) f(x) = f(c).
If all three conditions are met, we say that the function is continuous at x = c.
The continuity rule extends this concept to intervals. A function is continuous over an interval if it is continuous at every point within that interval.
There are three types of continuity:
1. Pointwise continuity: A function is continuous at a specific point.
2. Interval continuity: A function is continuous over a given interval.
3. Global continuity: A function is continuous over its entire domain.
The continuity rule provides us with a way to determine whether a function is continuous at a given point or over a given interval. It is an important tool in analyzing functions and understanding their behavior.
More Answers:
Understanding the Constant Function and Identity Function: Exploring Elementary Functions in MathematicsUnderstanding the Limit as x Approaches 0 of sinx/x: Exploring Trigonometric Properties and the Squeeze Theorem
Solving the Limit of (1 – cos(x))/x as x approaches 0 using Trigonometric Manipulation