Calculus AB · Unit 1: Limits and Continuity · 14 min read · Updated 2026-05-10
Defining continuity at a point — AP Calculus AB
AP Calculus AB · Unit 1: Limits and Continuity · 14 min read
1. Definition of Continuity at an Interior Point★★☆☆☆⏱ 4 min
Informally, a function is continuous at a point $x=a$ if you can draw the graph of the function through $x=a$ without lifting your pencil off the paper. Continuity is a foundational property that underpins nearly all advanced results in calculus, from the Intermediate Value Theorem to derivatives and the Fundamental Theorem of Calculus.
\lim_{x \to a} f(x) = f(a)
Each condition serves a distinct purpose: the first confirms we have a function value to compare, the second confirms the graph approaches the same value from both sides, and the third confirms the point actually lies on the approaching curve.
Exam tip: On AP FRQs, always explicitly check all three conditions when justifying continuity. Skipping any condition will cost you a point, even if your final conclusion is correct.
2. One-Sided Continuity for Domain Endpoints★★☆☆☆⏱ 3 min
For functions defined on closed intervals $[a,b]$, a two-sided limit cannot exist at the endpoints, since the function is not defined outside the interval. For this reason, we use one-sided continuity to define continuity at endpoints.
A function is continuous on the full closed interval $[a,b]$ if it is continuous at all interior points, right-continuous at the left endpoint, and left-continuous at the right endpoint.
Exam tip: Never check for a two-sided limit at a domain endpoint. AP exam graders will penalize you for referencing a non-existent limit outside the function's domain.
3. Checking Continuity at Piecewise Function Boundaries★★★☆☆⏱ 4 min
The most common AP exam question on this topic asks you to check continuity at the boundary point where the definition of a piecewise function changes. Each individual piece of a piecewise function is almost always continuous (polynomials, rationals, trig functions are continuous on their domains), so the only possible discontinuity is at the boundary between pieces.
Find $f(a)$ from the piece that includes $x=a$.
Calculate the left-hand limit $\lim_{x \to a^-} f(x)$ using the piece defined for $x < a$.
Calculate the right-hand limit $\lim_{x \to a^+} f(x)$ using the piece defined for $x > a$.
Confirm both one-sided limits are equal (so the two-sided limit exists), then confirm the limit equals $f(a)$.
Exam tip: Always match the direction of the limit to the inequality of the piece. Swapping left and right pieces is the most common mistake on these problems.
4. AP-Style Worked Practice Problems★★★☆☆⏱ 3 min
Common Pitfalls
Why: Students confuse the unsimplified algebraic expression with the given function's definition, forgetting the function explicitly defines $f(3)$.
Why: Students forget that $f(a)$ could be undefined, or could be a different value than the limit.
Why: Students confuse two-sided continuity for interior points with the definition of continuity at endpoints.
Why: Students rush and misread the inequality signs defining the pieces.
Why: Students mix up one-sided continuity for endpoints with the requirements for interior points.