Continuity of Composite Functions

What Continuity of Composite Functions Means

A composite function is a function built by putting one function inside another. In notation, we write it as .

For example, is a composite function. The outside function is , while the inside function is .

The important question is: if the inside and outside functions are continuous, is the composite function also continuous?

Usually, yes. In fact, this is one of the most useful continuity facts in AP Calculus:

In words: if is continuous at , and is continuous at , then is continuous at .

The Continuity Theorem for Composite Functions

Suppose and . If:

is continuous at , and
is continuous at ,

then the composite function is continuous at .

This theorem is closely related to the limit theorem for composite functions. If the inside function approaches the correct value and the outside function is continuous there, you can essentially "pass the limit through" the outside function.

Why This Works

Remember the definition of continuity. A function is continuous at when:

Now suppose . If is continuous at , then:

And if is continuous at , then:

Putting those ideas together gives:

That is exactly the condition required for to be continuous at .

A Step-by-Step Method

When an AP Calculus problem asks you to determine whether a composite function is continuous, use this simple process:

Identify the inside function. Usually this is the expression being substituted into another function.
Identify the outside function. This is the function being evaluated at the inside expression.
Check where the inside function is continuous. Polynomials are continuous everywhere, while rational and radical functions may have restrictions.
Check the outside function at the relevant input. The important input is the value produced by the inside function.
Apply the composition theorem. If both continuity conditions are satisfied, the composite function is continuous.

Worked Examples

Example 1: Polynomial Inside a Polynomial

Determine whether is continuous at .

Step 1: Identify the inside function: .

Step 2: Since is a polynomial, it is continuous for every real number. Therefore, it is continuous at .

Step 3: The outside function is , which is also a polynomial and therefore continuous everywhere.

Step 4: The inside function produces . The outside function is continuous at that value.

Result: By the continuity theorem for composite functions, is continuous at .

Example 2: Square Root of a Composite Function

Determine the interval on which is continuous.

Step 1: The inside function is . Since it is a polynomial, it is continuous everywhere.

Step 2: The outside function is . The square-root function is continuous whenever its input is nonnegative.

Step 3: Require the inside function to be nonnegative: .

Step 4: Solve the inequality: .

Result: The composite function is continuous on .

Example 3: Rational Function Inside a Polynomial

Determine where is continuous.

Step 1: The inside function is .

Step 2: The denominator cannot equal zero, so . Therefore, the inside function is continuous everywhere except .

Step 3: The outside function is , a polynomial, so it is continuous for every real input.

Result: The composite function is continuous everywhere in its domain: .

Example 4: A Composition Involving a Logarithm

Determine where is continuous.

Step 1: The inside function is , which is continuous everywhere.

Step 2: The outside function is . The natural logarithm is continuous only when its input is positive.

Step 3: Therefore, require .

Step 4: Solve: .

Result: The composite function is continuous on .

Continuity Does Not Mean "Continuous Everywhere"

This distinction matters on the AP exam. A composite function can be continuous at every point in its domain without being continuous over all real numbers.

For example, consider:

The square-root function is continuous on its domain, but its domain is restricted by the requirement that:

So the correct conclusion is not "the function is continuous everywhere." The correct statement is that it is continuous on its domain, which in this case is .

Using Continuity to Evaluate Limits

The continuity theorem is especially useful when an AP Calculus question asks you to evaluate a limit involving a composite function.

If the relevant function is continuous at the point, you can often substitute directly.

Example 5: Evaluating a Limit by Continuity

Evaluate .

Step 1: Recognize that is a polynomial, so it is continuous everywhere.

Step 2: Therefore, substitute directly into the function.

Step 3: .

Result: .

Example 6: A Composite Trigonometric Function

Evaluate .

Step 1: The inside function is continuous everywhere.

Step 2: The sine function is continuous for every real input.

Step 3: Therefore, the composite function is continuous everywhere and direct substitution is valid.

Result: .

Common Mistakes

Confusing the inside function with the entire composite function

In , the inside function is , not the entire expression. Identifying the two layers correctly is the first step in applying the composition theorem.

Forgetting the domain of the outside function

Saying "both functions are continuous" is not enough when the outside function has restrictions. For example, is continuous only when its input is positive. Always check whether the inside function produces an allowable input.

Saying a function is continuous everywhere when it is only continuous on its domain

Rational functions, square-root functions, logarithms, and inverse trigonometric functions can have restricted domains. The safest AP Calculus statement is often "continuous on its domain."

Assuming composition automatically fixes a discontinuity

The composition theorem requires the relevant continuity conditions to be satisfied. If the outside function is discontinuous at the value produced by the inside function, you cannot automatically conclude that the composite is continuous.

Practice Problems

Question 1

Determine whether is continuous at .

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Question 2

Determine the interval on which is continuous.

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Question 3

Determine where is continuous.

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Question 4

Evaluate using continuity.

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