AP Calculus BC Unit 1 Topic 3: Definitions of continuity of a function at a point and over a domain

What to Know

Check the conditions of a theorem or method before applying it.
Show the setup before the calculation.
Interpret the result in context, including units when the problem supplies them.
Always connect this topic back to the larger unit: Limits and Continuity.

Detailed Notes

Definitions of continuity of a function at a point and over a domain should be studied through multiple representations. A graph may show behavior quickly, an equation may make calculation possible, and a verbal interpretation explains what the result means.

In AP Calculus BC, AP questions often award credit for setup and reasoning, not just final answers. Write the expression, theorem, condition, or model before doing the computation.

When this topic appears in free response, check whether the question asks for a value, a rate, an interval, a comparison, or a justification. Use units and context to make the final answer precise.

Key Vocabulary

Limit

The value a function approaches as the input approaches a specified value.

Continuity

A property of a function with no break, hole, or jump at a point or on an interval.

Asymptote

A line that a graph approaches but may not reach.

Intermediate Value Theorem

A theorem guaranteeing that continuous functions take on every value between two function values.

Quick Practice

How would you explain Definitions of continuity of a function at a point and over a domain in one or two AP-style sentences?

Name the concept, apply it to a specific example or source, and explain the reasoning that connects the evidence to your answer.