Confirming Continuity Over an Interval

What It Means to Be Continuous Over an Interval

A function is continuous over an interval when there are no breaks, holes, jumps, or vertical asymptotes anywhere in that interval. But on the AP Calculus exam, you need to be precise about what happens at the interior points and at the endpoints.

For a function to be continuous at an interior point , three conditions must hold:

exists.
exists.
.

When checking continuity over an entire interval, the key idea is that these conditions must hold at every point in the interval.

Continuity on a Closed Interval

Suppose we want to determine whether is continuous on the closed interval .

For every interior point, we use ordinary two-sided continuity. At the endpoints, however, we only need continuity from within the interval.

So, for continuity on :

At every , must hold.
At the left endpoint , the right-hand limit must equal .
At the right endpoint , the left-hand limit must equal .

This endpoint detail is important. A function does not need to have values approaching the endpoint from outside the interval because those points are not part of the interval being considered.

A Reliable Step-by-Step Method

Identify the interval. Determine whether it is open, closed, or half-open.
Identify possible discontinuities. Look for denominators equal to zero, even roots, logarithms, piecewise definitions, or other restrictions.
Check the domain. The function must be defined at every point where continuity is being claimed.
Check interior points. At an interior point, verify that the function is defined and that the two-sided limit equals the function value.
Check endpoints appropriately. Use a right-hand limit at the left endpoint and a left-hand limit at the right endpoint.
State the conclusion clearly. Explain why the function is continuous on the entire interval rather than simply saying "yes."

Worked Examples

Example 1: Polynomial Function

Determine whether is continuous on .

Step 1: Identify the function type. is a polynomial.

Step 2: Recall the continuity rule: every polynomial is continuous for all real numbers.

Step 3: Since is continuous everywhere, it is continuous at every point in .

Conclusion: is continuous on the entire interval.

Example 2: Rational Function

Determine whether is continuous on .

Step 1: A rational function is continuous wherever its denominator is nonzero.

Step 2: Set the denominator equal to zero:

Step 3: The only possible discontinuity is at .

Step 4: Because lies outside the interval , there are no discontinuities within the interval.

Conclusion: is continuous on .

Example 3: Rational Function That Is Not Continuous

Determine whether is continuous on .

Step 1: Find where the denominator is zero:

Step 2: The function is undefined at .

Step 3: Since lies inside the interval , the function is not defined at every point of the interval.

Conclusion: is not continuous on the interval.

Example 4: Piecewise Function

Determine whether the function below is continuous on :

Step 1: Each individual formula is continuous on the interval where it is used. The only point that requires special attention is the piecewise boundary, .

Step 2: Evaluate the function at the boundary:

Step 3: Evaluate the left-hand limit:

Step 4: Evaluate the right-hand limit:

Step 5: All three values agree, so the function is continuous at the boundary.

Conclusion: The function is continuous on the entire interval.

Why the Endpoints Are Different

One of the most common AP Calculus mistakes is treating an endpoint exactly like an interior point.

Suppose a function is defined on . At the interior point , values of can approach the point from both sides.

But at the left endpoint , there are no points in the interval to the left. Therefore, we use only the right-hand limit:

Likewise, at the right endpoint , we use only the left-hand limit:

This is why a function can be continuous on a closed interval even though its behavior outside that interval is irrelevant.

Using Known Continuity Rules

Many AP Calculus questions can be solved without calculating a single limit. Instead, recognize functions that are already known to be continuous.

Polynomials are continuous for all real numbers.
Rational functions are continuous wherever the denominator is nonzero.
Exponential functions are continuous for all real numbers.
Sine and cosine are continuous for all real numbers.
Logarithmic functions are continuous wherever their arguments are positive.
Root functions are continuous wherever their domains are defined.
Combinations and compositions of continuous functions are continuous wherever they are defined.

For example, consider:

You do not need to test every point individually. The expression is a combination of continuous functions, and its denominator is nonzero throughout the specified interval. Therefore, continuity follows from the continuity rules.

Common Mistakes

Checking only the formula and ignoring the domain

A rational function may look perfectly ordinary, but it is not continuous where its denominator equals zero. Always check whether the function is actually defined throughout the interval.

Checking an entire interval by testing a few points

Testing continuity at three or four points does not prove continuity over an interval. Instead, use known continuity rules or identify every possible point of discontinuity.

Using a two-sided limit at an endpoint

For a closed interval, use a right-hand limit at the left endpoint and a left-hand limit at the right endpoint. There is no need to consider values outside the interval.

Forgetting the piecewise boundary

Each piece may be continuous on its own, but the entire function can still be discontinuous where the pieces meet. Always check the transition point.

Saying only "the graph has no breaks"

That may describe the intuition, but AP Calculus often expects mathematical justification. Use the function's domain, known continuity rules, or the appropriate limit conditions to support your conclusion.

AP Calculus Exam Strategy

When an AP question asks you to confirm that a function is continuous on an interval, don't automatically start calculating limits at every point. First look for a faster structural argument.

Identify the type of function.
Find all possible restrictions on its domain.
Determine whether any restriction occurs inside the requested interval.
If the function is piecewise, check every boundary between pieces.
Handle endpoints using one-sided continuity when necessary.
Write a conclusion that explicitly connects your reasoning to the interval.

A strong AP-style conclusion might look like this:

The important phrase is "for every in the interval". Continuity over an interval is a statement about all points in that interval, not just one particular value.

Practice Problems

Question 1

Is continuous on ?

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Question 2

Is continuous on ?

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Question 3

Is continuous on ?

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Question 4 (Piecewise)

Determine whether the following function is continuous on :

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