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Continuity

Continuity

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The Concept of Continuity: An Intuitive Understanding

Before delving into the technical definition of continuity, let's first establish an intuitive understanding of continuous functions. Perhaps you have heard the phrase "a function is continuous if you can draw it without lifting your pencil". This statement suggests that there are certain conditions that must be met for a function to be continuous, so let's explore what those conditions are.

Let's begin by considering a single point p. If we attempt to draw a function at p without lifting our pencil, we may encounter difficulties due to a hole or undefined point at that location. Therefore, we must assume that the function is defined at p (denoted as "assume exists") to be able to draw it continuously.

The Definition of Continuity

Having a defined function at p is not enough to ensure continuity. We must also confirm that the limit at p from the left and the right are equal. For instance, a function may have a defined value at p, but the left and right limits are not equal, indicating that the limit as x approaches p does not exist. Therefore, for a function to be considered continuous, the limit must also exist.

Combining these ideas, we can formulate a definition of continuity:

Defining Continuity

A function is said to be continuous at a point p if and only if the following three conditions are satisfied:

  1. The function is defined at p.
  2. The limit from the left and right is equal.
  3. The function value at p is equal to the limit.

If any of these conditions are not met, the function is considered discontinuous at p.

The phrase "if and only if" is used to indicate a biconditional logic statement, meaning if A is true, then B is true and vice versa.

Checklist for Continuity

Utilizing this definition, we can establish a step-by-step procedure to determine if a function is continuous at a given point p:

  1. Confirm that the function is defined at p.
  2. Check if the limit exists at p.
  3. Compare the limit to the function value at p.
  4. If any of these conditions are not met, the function is not continuous at p.

Note that the terms "discontinuous at a point" and "has a discontinuity at that point" can be used interchangeably.

Examples of Continuous Functions

Now, let's apply this definition to some functions and determine their continuity at a specific point p.

Example 1:

Is the function continuous at p = 2?

Answer: When we attempt to evaluate the function at p = 2, we encounter division by zero, indicating that the function is not defined at that point. Graphing the function also confirms this with a vertical asymptote at p = 2.

Example 2:

Does the function have continuity at x = 2?

Answer: While the function is defined at x = 2, the limit does not exist as the right and left limits are not equal. Therefore, the function is not continuous at x = 2.

Example 3:

Let's modify the previous function to and examine its continuity at x = 2.

Answer: Now, the limit as x approaches 2 is 2, but the function value at x = 2 is 4. Thus, the function is still not continuous at x = 2.

Example 4:

Determine if the function is continuous when x = 3.

Answer: The key is to pay attention to the given point, which in this case is x = 3. While the function changes definition at x = 2, it is not relevant for this question. We can observe that the function is not continuous at x = 3 because it is not defined at that point.

Understanding Continuity in Calculus

In calculus, it is crucial to determine if a function is continuous at a specific point. The process is straightforward and involves checking if the function is defined at that point, if the limit exists, and if the limit is equal to the function value. Let's walk through the steps to determine continuity at a given point.

Step 1: Confirm Function Definition

The first step is to ensure that the function is defined at the designated point. This means that the function must have a value at that point. If the function is not defined, then it is not continuous at that point.

For example, for the function f(x) = 2x + 3, continuity at x = 2 can be verified as f(2) = 7.

Step 2: Check for Limit Existence

The next step is to verify if the limit exists at the given point. If the limit does not exist, then the function is not considered continuous at that point.

For instance, for the function f(x) = 1/x, the limit as x → 0 does not exist, indicating that the function is not continuous at x = 0.

In calculus, determining the continuity at a given point means that the limit from the left must be equal to the limit from the right. In simpler terms, both one-sided limits must exist and be equal to each other.

For instance, let's consider the function f(x) = 3x + 1 and the point x = 2. The limit from the left of x = 2 is 7, while the limit from the right is also 7, meaning the limit exists and is equal from both sides.

Step 3: Checking the Equality of Limit and Function Value

The next step is to check if the limit and the function value at the given point are equal. If they are, then the function is continuous at that point.

For our example, both the limit and the function value at x = 2 are 7, therefore the function is continuous at this point.

Determining Continuity at a Different Point

Now, let's apply these steps to a different point. Consider the function g(x) = x^2 + 1 and the point x = 3.

Step 1: Checking if the Function is Defined

To begin, we must check if the function is defined at x = 3. And it is, because g(3) = 10.

Step 2: Checking if the Limit Exists

Next, we need to find the limit from the left and the right of x = 3:

  • Limit from the left: g(3) = 10
  • Limit from the right: g(3) = 10

Since both limits are equal, the limit exists from both sides.

Step 3: Checking the Equality of Limit and Function Value

The limit is 10 and the function value at x = 3 is also 10, indicating that the limit equals the function value. Hence, the function is continuous at this point.

What if the Function is Not Continuous?

If the function is not continuous at a given point, it means there is a gap or jump at that point. In other words, the function is defined, but the limit is not equal to the function value at that point.

For example, let's look at the function h(x) = 1/x and the point x = 0. By following the same steps, we can determine that the function is not continuous at this point because the limit from the left is -∞ and the limit from the right is +∞.

Why is Continuity Important?

The concept of continuity is crucial in calculus because it helps us understand the behavior of a function at a specific point. If a function is not continuous at a point, it can indicate a sudden change or discontinuity in the data being modeled.

Moreover, understanding continuity is beneficial in identifying the properties and behaviors of different functions over an interval or the real line.

In Conclusion

In summary, continuity in calculus refers to the smoothness of a function at a specific point. It is determined by checking if the function is defined, if the limit exists, and if the limit equals the function value at that point. The understanding of continuity allows for a better interpretation and analysis of functions and their behavior over intervals or the whole real line.

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