1.11 Defining Continuity at a Point

4 min read•june 18, 2024

AP Calculus AB/BC ♾️

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In this guide, we will discuss the concept of continuity at a point in Calculus, which is a fundamental concept that builds upon the understanding of limits and discontinuities. Continuity at a point refers to the property of a function where the function's value and its limit at that point are equal.

How to Determine Continuity at a Point

To determine continuity at a point, we use the formal definition of continuity: a function f(x) is continuous at a point c if and only if the following three conditions are met:

The limit of the function as x approaches c exists.
The function is defined at the point c.
The limit of the function as x approaches c is equal to the function's value at c.

This definition is often written in mathematical notation as: lim x->c f(x) = f(c)

It's important to note that this definition applies to both real and complex functions, even though in real functions, it's possible to graph the function and observe the continuity or discontinuity.

Examples

Now, let's look at some examples to help illustrate the concept of continuity at a point:

Example 1: Is the function f(x) = x^2 continuous at x = 2? Solution:

The limit of f(x) as x approaches 2 exists and is equal to 4.
The function is defined at x = 2.
The limit of f(x) as x approaches 2 is equal to the function's value at x = 2, which is 4. Therefore, the function f(x) = x^2 is continuous at x = 2.

Example 2: Is the function f(x) = 1/x continuous at x = 0? Solution:

The limit of f(x) as x approaches 0 does not exist.
The function is defined at x = 0.
The limit of f(x) as x approaches 0 is not equal to the function's value at x = 0, which is undefined. Therefore, the function f(x) = 1/x is not continuous at x = 0.

Example 3: Is the function f(x) = sqrt(x) continuous at x = 4? Solution:

The limit of f(x) as x approaches 4 exists and is equal to 2.
The function is defined at x = 4.
The limit of f(x) as x approaches 4 is equal to the function's value at x = 4, which is 2. Therefore, the function f(x) = sqrt(x) is continuous at x = 4.

Example 4: Is the function f(x) = |x| continuous at x = 0? Solution:

The limit of f(x) as x approaches 0 from the left is -0 and from the right is 0.
The function is defined at x = 0.
The limit of f(x) as x approaches 0 from both sides is not equal to the function's value at x = 0, which is 0. Therefore, the function f(x) = |x| is not continuous at x = 0.

Example 5: Is the function f(x) = x^3/(x-1) continuous at x = 1? Solution:

The limit of f(x) as x approaches 1 exists and is equal to 1.
The function is not defined at x = 1.

3. The limit of f(x) as x approaches 1 is not equal to the function's value at x = 1 which is undefined. Therefore, the function f(x) = x^3/(x-1) is not continuous at x = 1.

Example 6: Is the function f(x) = sin(1/x) continuous at x = 0? Solution:

The limit of f(x) as x approaches 0 does not exist, as the function oscillates between positive and negative values.
The function is defined at x = 0
The limit of f(x) as x approaches 0 is not equal to the function's value at x = 0, which is 0. Therefore, the function f(x) = sin(1/x) is not continuous at x = 0.

Example 7: Is the function f(x) = x^2*e^x continuous at x = 0? Solution:

The limit of f(x) as x approaches 0 exists and is equal to 0.
The function is defined at x = 0
The limit of f(x) as x approaches 0 is equal to the function's value at x = 0, which is 0. Therefore, the function f(x) = x^2*e^x is continuous at x = 0.

In conclusion, continuity at a point is an essential concept in Calculus that builds upon our understanding of limits and discontinuities. To determine continuity at a point, we use the formal definition that a function is continuous at a point c if and only if the limit of the function as x approaches c exists, the function is defined at the point c, and the limit of the function as x approaches c is equal to the function's value at c. By understanding and being able to apply this definition, as well as by practicing with examples like the ones provided, you will be well-prepared to succeed in your studies of Calculus.

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👑Unit 1 – Limits & Continuity

1.0Unit 1 Overview: Limits and Continuity

1.1Introducing Calculus: Can Change Occur at An Instant?

1.2Defining Limits and Using Limit Notation

1.3Estimating Limit Values from Graphs

1.4Estimating Limit Values from Tables

1.5Determining Limits Using Algebraic Properties of Limits

1.6Determining Limits Using Algebraic Manipulation

1.7Selecting Procedures for Determining Limits

1.8Determining Limits Using the Squeeze Theorem

1.9Connecting Multiple Representations of Limits