Understanding limits and continuity unlocks many topics in AP® Calculus. When limits behave nicely, derivatives and integrals often follow quickly. However, graphs sometimes feature a break, called a discontinuity. Correctly naming the break saves time on free-response questions. This post reviews the three official types of discontinuity—removable, jump, and infinite—and shows how limits classify each one. Therefore, by the end of the lesson, spotting a discontinuity will feel almost automatic.
What We Review
What Is a Discontinuity?
A function is continuous at a point if its graph has no gaps there. Informally, a continuous curve can be drawn without lifting the pencil. A discontinuity is any place where that smooth drawing fails—a hole, a sudden jump, or an unbounded spike. Limits provide the test: if the limit behaves strangely at a point, the graph is broken there.
The Three Main Types of Discontinuity
Removable Discontinuity
1. Definition: A removable discontinuity is a single missing or mismatched point in an otherwise smooth graph.
2. Limit test: The two-sided limit exists and is finite, yet the function value is either undefined or different from that limit.
3. Step-by-Step Example (removable discontinuity examples)
Consider
f(x)=\dfrac{x^2-4}{x-2}- Factor the numerator.
- x^2-4=(x-2)(x+2)
- Cancel common factors for all x\neq 2.
- f(x)=x+2,\quad x\neq 2
- Limit computation. \displaystyle\lim_{x\to 2}f(x)=\lim_{x\to 2}(x+2)=4 Therefore, the limit exists.
- Function value check.
- Because the original denominator is zero at x=2, f(2) is undefined. A hole sits at the point (2, 4).
- Graph sketch explanation
- The line y=x+2 would be continuous, but a tiny open circle at (2, 4) marks the gap.
- How to remove it
- Define a new function g(x)=\begin{cases}f(x) & x\neq 2\\4 & x=2\end{cases}.
- Now g(x) is continuous everywhere.
Jump Discontinuity
1. Definition: A jump discontinuity occurs when the graph suddenly leaps from one height to another.
2. Limit test: The left-hand and right-hand limits both exist and are finite, yet they are not equal. Consequently, the two-sided limit fails.
3. Step-by-Step Example (jump discontinuity example)
Let
h(x)=\begin{cases}x-1,& x<1\\2-x,& x\ge 1\end{cases}- Left-hand limit. \displaystyle\lim_{x\to 1^-}h(x)=1-1=0
- Right-hand limit. \displaystyle\lim_{x\to 1^+}h(x)=2-1=1 Because 0 ≠ 1, the two-sided limit does not exist, so there is a jump at x=1.
- Graph note
- One branch approaches (1, 0) from the left; the other starts at (1, 1). The vertical gap between those y-values illustrates the jump.
Infinite Discontinuity (Vertical Asymptote)
1. Definition: An infinite discontinuity happens when the function heads toward positive or negative infinity near a point, producing a vertical asymptote.
2. Limit test: At least one one-sided limit evaluates to \pm\infty.
3. Step-by-Step Example (infinite discontinuity example)
Take
p(x)=\dfrac{3}{(x-2)^2}- One-sided limits. \displaystyle\lim_{x\to 2^-}p(x)=\infty,\qquad \lim_{x\to 2^+}p(x)=\infty The function grows without bound on both sides; therefore, x=2 is a vertical asymptote.
- Graph insight
- Both branches rise steeply near the line x=2, forming the classic “wall” of an infinite discontinuity.
How to Spot Discontinuities on a Graph
When approaching a graph, use this checklist:
- Look for open circles—those signal removable discontinuities.
- Next, watch for sudden vertical gaps where the curve jumps to a new height; that indicates a jump discontinuity.
- Finally, observe any vertical lines that the curve approaches but never touches. Those vertical asymptotes reveal infinite discontinuities.
Quick Reference Chart: Vocabulary & Key Facts
| Term | Meaning | Limit Behavior |
| Limit | Predicted y-value as x approaches a point | Finite number or ±∞ |
| Continuity | No break at a point | Limit equals function value |
| Removable discontinuity | Single missing or wrong point | Finite limit exists |
| Jump discontinuity | Sudden change in height | Left and right limits unequal |
| Infinite discontinuity | Graph blows up vertically | Limit = ±∞ |
| Vertical asymptote | x-value where function is unbounded | Same as infinite discontinuity |
| One-sided limit | Limit from left or right only | Used to detect jumps and asymptotes |
Conclusion
Recognizing the three types of discontinuity—removable, jump, and infinite—simplifies limit questions and speeds up AP® Calculus solutions. Correct classification highlights whether a hole can be patched, a step ignored, or an asymptote noted. Therefore, always check both the algebraic expression and the graph. Regular practice with these tests will make each type of discontinuity instantly recognizable on exam day.
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