Identifying the Discontinuities

Oct 13, 2025 by windailys.com 82 views

Introduction

In mathematics, a piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain. These sub-functions are often referred to as "pieces" of the function. In this article, we will explore the concept of completing the definition of a piecewise function to ensure continuity over its domain.

What is Continuity?

Continuity is a fundamental concept in calculus that refers to the ability of a function to be drawn without lifting the pencil from the paper. In other words, a function is continuous if it can be drawn in a single, unbroken motion. Mathematically, a function f(x) is continuous at a point x = a if the following conditions are met:

The function is defined at x = a.
The limit of the function as x approaches a exists.
The limit of the function as x approaches a is equal to the value of the function at x = a.

The Piecewise Function

The piecewise function h(x) is defined as:

${ h(x) = \begin{cases} x^3, & x \ \textless \ 0 \\ a, & x = 0 \\ \sqrt{x}, & 0 \ \textless \ x \ \textless \ 4 \\ b, & x = 4 \\ 4 - \frac{1}{2} x, & x \ \textgreater \ 4 \\ \end{cases} }$

Identifying the Discontinuities

To complete the definition of the piecewise function, we need to identify the points where the function is discontinuous. These points are called "discontinuities." In this case, the function is discontinuous at x = 0 and x = 4.

Completing the Definition at x = 0

At x = 0, the function is defined as a. However, the limit of the function as x approaches 0 from the left is not equal to a. In fact, the limit of x^3 as x approaches 0 from the left is 0. Therefore, to complete the definition of the function at x = 0, we need to set a = 0.

Completing the Definition at x = 4

At x = 4, the function is defined as b. However, the limit of the function as x approaches 4 from the right is not equal to b. In fact, the limit of 4 - (1/2)x as x approaches 4 from the right is 0. Therefore, to complete the definition of the function at x = 4, we need to set b = 0.

The Completed Piecewise Function

The completed piecewise function is:

${ h(x) = \begin{cases} x^3, & x \ \textless \ 0 \\ 0, & x = 0 \\ \sqrt{x}, & 0 \ \textless \ x \ \textless \ 4 \\ 0, & x = 4 \\ 4 - \frac{1}{2} x, & x \ \textgreater \ 4 \\ \end{cases} }$

Conclusion

In this article, we have completed the definition of a piecewise function to ensure continuity over its domain. We have identified the points of discontinuity and completed the definition of the function at these points. The completed piecewise function is a continuous function that can be drawn without lifting the pencil from the paper.

Example Problems

Find the limit of the function f(x) = x^2 as x approaches 2 from the right.
Find the limit of the function f(x) = 1/x as x approaches 0 from the left.
Find the limit of the function f(x) = sin(x) as x approaches π/2 from the right.

Solutions

The limit of the function f(x) = x^2 as x approaches 2 from the right is 4.
The limit of the function f(x) = 1/x as x approaches 0 from the left is -∞.
The limit of the function f(x) = sin(x) as x approaches π/2 from the right is 1.

Practice Problems

Find the limit of the function f(x) = x^3 as x approaches 0 from the left.
Find the limit of the function f(x) = 1/x^2 as x approaches 0 from the right.
Find the limit of the function f(x) = sin(x) as x approaches 0 from the left.

Solutions

The limit of the function f(x) = x^3 as x approaches 0 from the left is 0.
The limit of the function f(x) = 1/x^2 as x approaches 0 from the right is ∞.
The limit of the function f(x) = sin(x) as x approaches 0 from the left is 0.
Q&A: Completing the Definition of a Piecewise Function for Continuity ====================================================================

Q: What is a piecewise function?

A: A piecewise function is a function that is defined by multiple sub-functions, each applied to a specific interval of the domain.

Q: What is continuity in mathematics?

A: Continuity is a fundamental concept in calculus that refers to the ability of a function to be drawn without lifting the pencil from the paper. In other words, a function is continuous if it can be drawn in a single, unbroken motion.

Q: How do I identify the points of discontinuity in a piecewise function?

A: To identify the points of discontinuity, you need to examine the function at each point where the sub-functions change. In the case of the piecewise function h(x), the points of discontinuity are x = 0 and x = 4.

Q: How do I complete the definition of a piecewise function at a point of discontinuity?

A: To complete the definition of a piecewise function at a point of discontinuity, you need to examine the limit of the function as x approaches the point from both the left and the right. If the limits are equal, then the function is continuous at that point. If the limits are not equal, then the function is discontinuous at that point.

Q: What is the limit of a function as x approaches a point from the left?

A: The limit of a function f(x) as x approaches a point a from the left is the value that the function approaches as x gets arbitrarily close to a from the left side.

Q: What is the limit of a function as x approaches a point from the right?

A: The limit of a function f(x) as x approaches a point a from the right is the value that the function approaches as x gets arbitrarily close to a from the right side.

Q: How do I find the limit of a function as x approaches a point from the left or right?

A: To find the limit of a function as x approaches a point from the left or right, you can use various techniques such as:

Direct substitution: If the function is defined at the point, then the limit is equal to the value of the function at that point.
Factoring: If the function can be factored, then you can cancel out the common factors to find the limit.
L'Hopital's rule: If the function is a ratio of two functions and the limit of the ratio is 0/0 or ∞/∞, then you can use L'Hopital's rule to find the limit.

Q: What is L'Hopital's rule?

A: L'Hopital's rule is a mathematical technique used to find the limit of a ratio of two functions when the limit of the ratio is 0/0 or ∞/∞. The rule states that if the limit of the ratio is 0/0 or ∞/∞, then the limit of the ratio is equal to the limit of the ratio of the derivatives of the two functions.

Q: How do I apply L'Hopital's rule?

A: To apply L'Hopital's rule, you need to follow these steps:

Check if the limit of the ratio is 0/0 or ∞/∞.
If the limit is 0/0 or ∞/∞, then differentiate the numerator and denominator separately.
Take the limit of the ratio of the derivatives.
If the limit of the ratio of the derivatives is still 0/0 or ∞/∞, then repeat the process until the limit is defined.

Q: What are some common applications of L'Hopital's rule?

A: L'Hopital's rule has many applications in mathematics, physics, and engineering. Some common applications include:

Finding the limit of a ratio of two functions when the limit of the ratio is 0/0 or ∞/∞.
Finding the derivative of a function using the definition of a derivative.
Finding the area under a curve using the definite integral.
Finding the volume of a solid using the definite integral.

Q: What are some common mistakes to avoid when applying L'Hopital's rule?

A: Some common mistakes to avoid when applying L'Hopital's rule include:

Not checking if the limit of the ratio is 0/0 or ∞/∞ before applying L'Hopital's rule.
Not differentiating the numerator and denominator separately.
Not taking the limit of the ratio of the derivatives.
Not repeating the process until the limit is defined.

Q: How do I practice applying L'Hopital's rule?

A: To practice applying L'Hopital's rule, you can try the following:

Start with simple examples and gradually move on to more complex examples.
Use online resources such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha to practice applying L'Hopital's rule.
Work with a study group or tutor to practice applying L'Hopital's rule.
Take online quizzes and tests to assess your understanding of L'Hopital's rule.