Calculus BC · AP CED Unit 1: Limits and Continuity · 14 min read · Updated 2026-05-10
Exploring types of discontinuities — AP Calculus BC
AP Calculus BC · AP CED Unit 1: Limits and Continuity · 14 min read
1. Overview of Discontinuity Classification★☆☆☆☆⏱ 2 min
Discontinuities occur when any of the three conditions for continuity at $x=a$ are violated: (1) $f(a)$ is defined, (2) $\lim_{x \to a} f(x)$ exists, (3) $\lim_{x \to a} f(x) = f(a)$. This topic requires you to systematically categorize the type of discontinuity, not just identify that a discontinuity exists.
Discontinuities are first split into removable (can be fixed by redefining one point) and non-removable (cannot be fixed by redefining one point). Non-removable discontinuities are further split into jump, infinite, and oscillating types. This topic makes up 4-6% of the AP Calculus BC exam, appearing in both multiple-choice and free-response questions.
2. Removable Discontinuities★★☆☆☆⏱ 3 min
Removable discontinuities most commonly occur in rational functions when a common factor cancels out from the numerator and denominator. They can also occur in piecewise functions where the defined point does not match the surrounding limit.
3. Non-Removable Discontinuities: Jump★★☆☆☆⏱ 3 min
Jump discontinuities get their name from the 'jump' the function makes from one value to another at the point $a$. Because both one-sided limits are finite but unequal, there is no way to choose a single value for $f(a)$ that will make the two-sided limit exist, hence the discontinuity is non-removable.
4. Non-Removable Discontinuities: Infinite★★☆☆☆⏱ 3 min
Infinite discontinuities occur when a factor of the denominator does not cancel with a matching factor in the numerator of a rational function. Because the limit is not finite, the discontinuity cannot be removed by redefining $f(a)$.
5. Non-Removable Discontinuities: Oscillating★★★☆☆⏱ 2 min
This is the least frequently tested type of discontinuity on the AP exam, but it is still examinable, most often in multiple-choice questions asking to identify the type from a description or graph.
Common Pitfalls
Why: Students associate zero denominator with vertical asymptotes without factoring and checking for common factors.
Why: Students confuse the existence of a defined $f(a)$ with the requirement that the two-sided limit must exist for a discontinuity to be removable.
Why: Students mistake the midpoint of oscillation for the limit.
Why: Students forget to check that one-sided limits are equal before classifying.
Why: Students overgeneralize the case of cancelled common factors.