Calculus BC · Limits and Continuity · 25 min read · Updated 2026-05-13
Defining continuity at a point — AP Calculus BC
AP Calculus BC · Limits and Continuity · 25 min read
1. The Three-Part Definition of Continuity★☆☆☆☆⏱ 10 min
Intuitively, a function is continuous at a point if you can draw its graph through that point without lifting your pencil. Formally, this intuition translates to three required conditions, which you will use for all continuity checks on the AP exam.
Exam tip: AP free-response questions require you to explicitly reference all three conditions when justifying continuity.
2. Testing Continuity Algebraically★★☆☆☆⏱ 15 min
Piecewise functions are the most common scenario for algebraic continuity checks, since you need to compare one-sided limits from each side of the boundary point to confirm the two-sided limit exists.
3. Classifying Types of Discontinuities★★☆☆☆⏱ 12 min
Any function that fails the continuity test at a point is discontinuous there. Discontinuities are grouped into removable and non-removable categories based on whether the two-sided limit exists.
**Removable**: Limit exists, discontinuity is caused by an undefined point or mismatched function value (appears as a hole in the graph)
**Jump**: Non-removable; one-sided limits exist but are not equal, graph jumps between two values
**Infinite**: Non-removable; one or both limits approach $\pm \infty$, usually from a vertical asymptote
**Oscillating**: Non-removable; function oscillates near the point with no defined limit
4. Continuity and AP Exam Applications★★☆☆☆⏱ 10 min
The definition of continuity at a point is the foundation for nearly all major theorems in AP Calculus. For example, the Intermediate Value Theorem (IVT) only applies to functions that are continuous at every point on a closed interval.
Common Pitfalls
Why: You still need to confirm the limit exists and matches the function value
Why: The two-sided limit does not exist for infinite discontinuities, so they cannot be removable
Why: Many piecewise functions are designed to be continuous at the boundary
Why: One-sided continuity at endpoints counts as continuous for interval continuity