AP Calculus AB / Limits and Continuity
Lesson
Continuity and the ways it breaks
Continuous means you can draw through the point without lifting for a jump or a gap in the approaching height.
Learning goals
- Check the three-part continuity test at x = a.
- Classify a discontinuity as removable, jump, or infinite when the picture is clear.
Explanation
A function is continuous at a if f(a) exists, the limit as x approaches a exists, and those two numbers are equal. Piecewise functions fail this when the pieces do not meet. Rational functions fail it where a denominator is zero and the canceling trick does not save you.
A removable discontinuity is a hole you could fill with one point. A jump has two different one-sided limits. An infinite discontinuity has a vertical asymptote. Intermediate Value Theorem ideas later depend on continuity on a closed interval: if a continuous function goes from negative to positive, it hit zero somewhere between.
Key terms
- Continuity at a point. The limit of f as x approaches a exists, f(a) exists, and the two are equal.
- Removable discontinuity. A break where the two-sided limit exists but does not match the function’s value (or the value is missing).
Common mistakes
- Calling every undefined point a vertical asymptote.
- Using the Intermediate Value Theorem on a function that jumps.
Practice
Original Marlow Works items. Check the answer explanation after you try.
What is lim (x→3) of (x² − 9) / (x − 3)?
Original Marlow Works item — not a College Board question.
Take your time—this is practice, not a test.
Answer explanation
f is 3x for x < 1 and 4 for x ≥ 1. Is f continuous at 1?
Answer. No. The left-hand limit is 3 and f(1) = 4, so the pieces do not meet.
A jump fails the requirement that the two-sided limit equal the value.
Related resources
External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.
Official / link only
AP Calculus AB course page
Official eight-unit framework and 2027 exam updates.
College Board · All rights reserved · accessed 2026-10-01
Open sourceReference only — do not copy
OpenStax College Algebra 2e
Algebra reference; current terms include noncommercial and AI-ingestion limits.
OpenStax · Restricted — see publisher page · accessed 2026-10-01
Open source