Defining Continuity at a Point

What Continuity Means

Informally, a function is continuous at a point if its graph has no break, jump, or hole at that point. You can imagine drawing the graph near the point without lifting your pencil from the paper.

But AP Calculus requires more than the visual idea. Continuity has a precise mathematical definition based on the function's value and its limit.

A function is continuous at if the following three conditions are all satisfied:

The important idea is that all three conditions must be true. If even one fails, the function is not continuous at that point.

The Three Conditions for Continuity

Condition 1: Exists

The function must actually be defined at the point. In other words, there must be a real number assigned to .

If is undefined, continuity is immediately impossible. A hole in the graph often occurs because the function is not defined at the point.

Condition 2: Exists

The values of the function must approach one specific finite number as approaches .

If the left-hand and right-hand limits disagree, the two-sided limit does not exist, so the function cannot be continuous there.

Condition 3: The Limit Equals the Function Value

Finally, the number that the function approaches must be exactly the number the function actually equals at the point.

This is the condition that connects the limit to the actual value of the function. A function can have a perfectly good limit at a point and still fail to be continuous if the function's assigned value is different.

The Continuity Checklist

On an AP Calculus problem, you can turn the definition into a simple checklist:

1. Is defined?
2. Does exist?
3. Does ?

If the answer to all three questions is yes, then is continuous at .

Continuity from a Graph

Graphs can often make continuity easier to recognize. At a point , look for three things:

• Is there a defined point on the graph at ?
• Do the left and right sides approach the same ?
• Does the actual function value match that ?

If the graph has a hole, jump, or other break at , the function is not continuous there.

Piecewise Functions

Piecewise functions are especially important for continuity questions because the formula can change at a particular point.

Consider:

Suppose we want to determine whether this function is continuous at .

Because the formula changes at , we need to compare the function value with the limits from the left and right.

Finding a Missing Constant for Continuity

AP Calculus frequently asks you to determine a value of a constant that makes a piecewise function continuous.

The strategy is straightforward: set the limit equal to the function value.

Continuity and One-Sided Limits

For a function to be continuous at an interior point, its two-sided limit must exist. That means the left-hand and right-hand limits must agree.

So if:

then the two-sided limit does not exist, and the function is not continuous at .

Where Continuity Matters

Continuity is not just a definition to memorize. It is the assumption behind several major results in calculus.

• Limits: If a function is continuous at , then its limit at can be found by direct substitution.
• Derivatives: If a function is differentiable at a point, it must also be continuous there.
• Intermediate Value Theorem: Continuity guarantees that a function takes every intermediate value between two function values.
• Extreme Value Theorem: A continuous function on a closed interval is guaranteed to attain an absolute maximum and minimum.

This is why continuity appears repeatedly throughout AP Calculus. It is one of the foundational assumptions that allows many of the theorems you will learn later to work.

Common Mistakes

Practice Problems

The Big Idea

When AP Calculus asks whether a function is continuous at , do not rely only on what the graph looks like. Use the formal definition.

Think of continuity as a three-part chain: the function exists, the limit exists, and they agree. If one link breaks, continuity fails.