Concepts of Continuity

Definition of Continuity at a Point

A function f(x) is said to be continuous at a point x = c if the following condition holds:

This means that the limit of f(x) as x approaches c exists and is equal to the value of the function at c.

Continuity at Endpoints

• Left Endpoint: f(x) is continuous at a left endpoint x = a if

• Right Endpoint: f(x) is continuous at a right endpoint x = b if

Continuity Test

• Evaluate

• Find (you may need to find both and )

• Compare the results: If the limit exists and equals , the function is continuous at c.

Combinations of Continuous Functions

If f and g are continuous at a point, then the following combinations are also continuous at that point:

• Sum:

• Difference:

• Product:

• Quotient: (provided )

• Root: (if is defined on an interval containing and is odd, or if is even)

• Composition:

Continuous Extension and Removable Discontinuity

• Continuous Extension: If a value is not continuous at , but we can define so that becomes continuous, this is called a removable discontinuity.

• Removable Discontinuity: exists, but or is undefined. We can "remove" the discontinuity by redefining to match the limit.

Intermediate Value Theorem (IVT)

If f is continuous on a closed interval and is between and , then there exists a value in such that .

Piecewise Functions and Continuity

Example: Analyzing a Piecewise Function

Given the function:

To determine continuity at specific points (e.g., , ), follow these steps:

1. Compute the left-hand and right-hand limits at the point.

2. Find the function value at the point (if defined).

3. Compare the limits and the function value to determine continuity.

Example: At :

• Left-hand limit:

• Right-hand limit:

• Since the left and right limits are not equal, is not continuous at .

Types of Discontinuities

• Removable Discontinuity: The limit exists, but the function value is either not defined or not equal to the limit. The discontinuity can be "removed" by redefining the function value.

• Non-removable Discontinuity: The limit does not exist (e.g., jump or infinite discontinuity), so the discontinuity cannot be removed by redefining the function value.

Determining Continuity of Functions

Common Function Types

• Polynomials: Continuous everywhere.

• Rational Functions: Continuous everywhere except where the denominator is zero.

• Root Functions: Continuous on their domains (for even roots, the radicand must be non-negative).

• Trigonometric, Exponential, and Logarithmic Functions: Continuous on their domains.

Examples

• is continuous for .

• is continuous for all real .

• is continuous for all ; at , the function is undefined, but the limit exists (), so a removable discontinuity exists at .

Table: Types of Discontinuities

Intermediate Value Theorem (IVT) Application

The IVT is often used to show that a function takes on every value between and on a closed interval if the function is continuous on that interval.

• Example: If and , then for any between -2 and 4, there exists in such that .

Summary of Steps to Test Continuity at a Point

1. Find and .

2. If both limits exist and are equal, find .

3. If , then is continuous at .

4. If not, determine the type of discontinuity (removable or non-removable).

Additional info: The worksheet also includes exercises for students to practice identifying points of continuity and discontinuity, and to classify discontinuities as removable or non-removable, as well as applications of the Intermediate Value Theorem.