Calculus AB · CED Unit 1: Limits and Continuity · 14 min read · Updated 2026-05-10
Exploring types of discontinuities — AP Calculus AB
AP Calculus AB · CED Unit 1: Limits and Continuity · 14 min read
1. Continuity and Discontinuity Basics★★☆☆☆⏱ 3 min
A function is discontinuous at $x=a$ if it fails any of the three conditions for continuity at $a$. Exploring types of discontinuities is the process of classifying discontinuities based on which condition fails, and the behavior of the function’s limits at that point. This topic makes up 10–12% of the AP Calculus AB exam weight, and is a prerequisite for topics like the Intermediate Value Theorem, differentiability, and integration.
2. Removable Discontinuities★★☆☆☆⏱ 4 min
Removable discontinuities most commonly occur in rational functions when a common linear factor cancels from the numerator and denominator, leaving a "hole" in the graph. They can also occur in piecewise functions when the defined function value at a breakpoint does not match the approaching limit.
3. Jump Discontinuities★★★☆☆⏱ 3 min
Jump discontinuities are most common in piecewise functions with different expressions on either side of a breakpoint, and in step functions like the floor or ceiling functions. The value of $f(a)$ (whether defined or not) does not change the classification, as long as the one-sided limits are finite and unequal.
4. Infinite Discontinuities★★★☆☆⏱ 4 min
For rational functions, infinite discontinuities occur when the denominator is zero at $x=a$, but the numerator is non-zero at $x=a$, so the fraction grows without bound as $x$ approaches $a$. If the factor causing the zero denominator does not cancel with any factor in the numerator, the discontinuity is infinite.
Common Pitfalls
Why: Students assume all undefined points are holes, forgetting that only discontinuities with a finite existing limit are removable.
Why: Students confuse the requirement of an existing finite two-sided limit for removable discontinuities with just the function being undefined.
Why: Students forget the other two continuity conditions: the limit must exist, and the limit must equal the function value.
Why: Students confuse 'one-sided limits exist' with 'two-sided limit exists'.
Why: Students do not factor rational functions completely, so they miss common factors that cancel to create a removable discontinuity.