In AP Calculus, a discontinuity is an x-value where a function fails the three-part continuity definition (f(c) exists, the limit exists, and they match). The AP CED names three types you must identify: removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes (EK LIM-2.A.1).
A discontinuity is any point where a function breaks the formal definition of continuity. Per EK LIM-2.A.2, a function f is continuous at x = c only if three things are true at once. First, f(c) actually exists. Second, the limit of f(x) as x approaches c exists. Third, those two values are equal. If even one condition fails, you have a discontinuity at x = c.
The CED (EK LIM-2.A.1) sorts discontinuities into three types, and which one you have depends on which condition failed. A removable discontinuity is a hole in the graph. The limit exists, but it doesn't match f(c), or f(c) doesn't exist at all. A jump discontinuity happens when the left-hand and right-hand limits both exist but disagree, so the graph leaps from one value to another. A discontinuity due to a vertical asymptote (sometimes called an infinite discontinuity) happens when the function blows up to infinity near x = c, so the limit doesn't exist as a real number. Quick mental model: a hole you could patch with one point, a step you can't cross, and a wall the graph never touches.
Discontinuities live in Unit 1: Limits and Continuity, specifically Topics 1.10 (Exploring Types of Discontinuities) and 1.11 (Defining Continuity at a Point). Both topics share the same learning objective (1.10.A and 1.11.A), which says you have to justify conclusions about continuity using the definition. That word "justify" matters. On the AP exam, saying "the graph has a break" earns nothing. You earn credit by checking the three conditions of EK LIM-2.A.2 and stating which one fails. Discontinuity is also the gatekeeper concept for the rest of the course, because continuity is a hypothesis for the Intermediate Value Theorem, differentiability, and the definite integral. If you can't spot and classify discontinuities, those later units get shaky fast.
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Continuous (Unit 1)
Discontinuity is literally defined as the failure of continuity, so the two terms are the same three-part checklist read in opposite directions. To prove continuity you verify all three conditions of EK LIM-2.A.2; to prove a discontinuity you find the one that breaks.
Removable Discontinuity (Unit 1)
This is the friendliest type, because the limit still exists. For rational functions, you can often factor and cancel the troublesome term, which is exactly what one classic exam-style question asks you to do. Removable holes are also why redefining f(c) to equal the limit can "fix" a function.
Asymptote (Units 1 and beyond)
A vertical asymptote is one of the three CED discontinuity types, and it's the unfixable one. The function runs off to infinity, so no single point value can patch it. Vertical asymptotes come back in Unit 6 with improper-looking integrands and in curve sketching.
Differentiability (Unit 2)
Differentiability implies continuity, so any discontinuity instantly kills the derivative at that point. The reverse fails, though. A function like f(x) = |x| is perfectly continuous at x = 0 but still not differentiable there, a distinction the exam loves to test.
Discontinuity shows up mostly in multiple-choice classification questions. Typical stems give you a piecewise function, a rational function, or a graph and ask which type of discontinuity exists at a point. Practice questions hit the standard patterns, like identifying that a "hole" is a removable discontinuity, recognizing that a vertical asymptote creates an infinite (non-removable) discontinuity, and knowing that you can factor and cancel a rational expression to remove a removable discontinuity. Watch for the trap question asking what discontinuity f(x) = |x| has at x = 0; the answer is none, since |x| is continuous there (just not differentiable). On free-response, no released FRQ uses the word "discontinuity" as its centerpiece, but continuity justifications are a recurring rubric point. When asked whether f is continuous at x = c, write out all three conditions from EK LIM-2.A.2 with values, then state your conclusion. That's what "justify using the definition" means.
Both often come from a zero in the denominator of a rational function, which is why they get mixed up. The difference is what happens after you factor. If the factor cancels, like (x-2)/(x²-4) at x = 2, the limit exists and you get a removable hole. If the factor doesn't cancel, like at x = -2 in that same function, the output blows up and you get a vertical asymptote. Cancel means hole, no cancel means asymptote.
A discontinuity occurs at x = c whenever the function fails the three-part definition of continuity, meaning f(c) doesn't exist, the limit doesn't exist, or the two values don't match.
The AP CED names exactly three types of discontinuities: removable (a hole), jump (left and right limits disagree), and discontinuities due to vertical asymptotes (EK LIM-2.A.1).
A removable discontinuity is the only kind where the limit still exists, and for rational functions you can usually expose it by factoring and canceling.
A jump discontinuity means both one-sided limits exist but are different, which is the signature behavior of many piecewise-defined functions.
To earn credit on the exam, justify continuity conclusions by explicitly checking f(c), the limit, and whether they're equal, not by describing the graph.
A corner, like f(x) = |x| at x = 0, is NOT a discontinuity; the function is continuous there but not differentiable.
A discontinuity is an x-value where a function fails the definition of continuity, meaning f(c) doesn't exist, the limit as x approaches c doesn't exist, or the limit doesn't equal f(c). The CED covers it in Topics 1.10 and 1.11 of Unit 1.
No. The absolute value function is continuous everywhere, including x = 0, because the limit equals f(0) = 0. It has a corner there, so it's not differentiable, but non-differentiable is not the same as discontinuous.
At a removable discontinuity the limit exists and the graph just has a hole, which you can often reveal by canceling a common factor in a rational expression. At a vertical asymptote the function goes to infinity, the limit doesn't exist as a real number, and nothing can fix it.
Per EK LIM-2.A.1, they are removable discontinuities (holes), jump discontinuities (one-sided limits exist but differ), and discontinuities due to vertical asymptotes (the function blows up to infinity).
Yes, and that's exactly what a removable discontinuity is. The limit exists, but either f(c) is undefined or f(c) doesn't equal the limit. Jump and asymptote discontinuities, by contrast, are points where the (two-sided) limit fails to exist.