Understanding Continuity Rules in Calculus | A Fundamental Concept Explained

Continuity Rule

In mathematics, the continuity rule is a fundamental concept in calculus that describes the behavior of functions

In mathematics, the continuity rule is a fundamental concept in calculus that describes the behavior of functions. It states that if a function f(x) is continuous at a point x = c, then certain properties hold true. Let’s break it down step by step:

1. Continuity at a point: A function f(x) is said to be continuous at a point c if the following conditions are satisfied:
a. The function is defined at c, meaning f(c) is defined.
b. The limit of the function as x approaches c exists, denoted as lim(x→c) f(x).
c. The limit of the function as x approaches c is equal to the value of the function at c, i.e., lim(x→c) f(x) = f(c).

2. Continuity rule at a single point: Let’s assume that functions g(x) and h(x) are continuous at x = c, and let k(x) be a function defined by k(x) = g(x) + h(x). Then, k(x) is also continuous at x = c. This means that the sum of two continuous functions is also continuous at the given point.

3. Continuity rule for the product of two functions: If g(x) and h(x) are continuous at x = c, then their product function k(x) = g(x) * h(x) is continuous at x = c. Consequently, the product of two continuous functions is continuous at the given point.

4. Continuity rule for the composition of two functions: If g(x) is continuous at x = c and h(x) is continuous at g(c), then the composite function k(x) = h(g(x)) is continuous at x = c. In simpler terms, if you have two continuous functions and you compose them by substituting one into the other, the resulting function remains continuous at the given point.

These are a few examples of continuity rules in calculus, and they allow us to analyze the behavior of functions and make deductions about their continuity. Continuity is a foundational concept in calculus, providing a bridge between algebraic manipulation and the geometric properties of functions.

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