How do holes, jumps and unbounded breaks differ?
You will be able to: Classify a discontinuity from its one-sided behavior.
BC foundation: Unit 1 shares its limits-and-continuity objectives with AB. If you have studied these ideas before, use the explanations and written challenges to check your reasoning. Later BC units build on these foundations; this unit does not require series or advanced integration.
How do holes, jumps and unbounded breaks differ?
A missing graph point, a price jump, and a curve rising without bound all interrupt continuity, but they cannot all be repaired in the same way.
A useful starting point: How do a formula, table and graph tell the same story? →
Words and symbols before equations
- Removable
- A finite common limit exists but the point is missing or mismatched.
- Jump
- Finite one-sided limits exist and differ.
- Infinite discontinuity
- At least one side grows without bound.
- Classification
- A description supported by side limits and point value.
What this picture assumes
Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. The hole and reciprocal-square examples are undefined at 1. The jump assigns f(1)=6. The finite window clips unbounded curves; it does not cap function values.
Read the picture in three steps
- Read the axes, coordinates and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- Both sides approach 3; fill f(1)=3 to repair.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
A removable break has a finite limit L. Assigning f(a)=L repairs continuity at that point because the nearby behavior already agrees.
At a jump, the two finite side limits differ. No one assigned value can match both; moving a single filled dot cannot repair it.
For an infinite discontinuity such as 1/x² at zero, nearby outputs are unbounded. No finite assigned value can make the function continuous there.
Oscillatory behavior such as sin(1/x) is another reason a limit may fail; the three named types are not an exhaustive list of all possible discontinuities.
A worked example, step by step
Classify y=x+2 with the point at x=1 removed, a graph with side limits 2 and 4 at 1, and y=1/(x−1)² at 1.
- The line has a finite nearby limit 3, so its hole is removable.
- The second has unequal finite side limits, so it has a jump.
- The reciprocal square grows without bound, so its break is infinite.
- Only the first can be repaired by assigning one finite point value.
Being undefined at one input does not tell you which kind of discontinuity occurs.
Are holes and jumps both repaired by filling a dot?
Compare with an explanation
No. A jump has incompatible side limits.
Predict. Change one thing. Explain.
Switch among the three models. Predict whether one finite point assignment could fix each break, and justify using the side behavior.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Both sides approach 3; fill f(1)=3 to repair.
Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. The hole and reciprocal-square examples are undefined at 1. The jump assigns f(1)=6. The finite window clips unbounded curves; it does not cap function values.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using nearby function values, graph coordinates, or the hypotheses of a limit law or theorem. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionExplain how to distinguish a removable break from a jump using side limits, and say whether either requires f(a) to be undefined.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: A removable break has equal finite side limits.
- 1 point: A jump has unequal finite side limits.
- 1 point: A removable break is fixed by setting f(a) to the limit; a jump is not.
- 1 point: Either can have an assigned f(a); the assignment alone does not classify the break.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What makes a break removable?
A finite common limit can replace the point value.
RECALL 2What defines a jump?
Unequal finite one-sided limits.
RECALL 3Do these three types exhaust all possibilities?
No; oscillatory failures also occur.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do holes, jumps and unbounded breaks differ?
- Classify using nearby behavior first, then compare the assigned value.
Remember: Being undefined at one input does not tell you which kind of discontinuity occurs.
Conditions: Original equation-driven model; numeric readouts are rounded. Each +1 in the approach zoom divides the distance by 10; each +1 in the input scale multiplies the magnitude by 10. Graphs use labeled linear axes and finite sampled windows; exact claims require the lesson’s algebra or theorem. The hole and reciprocal-square examples are undefined at 1. The jump assigns f(1)=6. The finite window clips unbounded curves; it does not cap function values.
Refresh Kid · AP Calculus BC Unit 1 · Objectives LIM-2.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.10, LIM-2.A. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 1 has 16 official topics. Focused lesson titles and questions are original Refresh Kid teaching material. Topics 1.7 and 1.9 integrate earlier objectives and skills rather than introducing new numbered learning objectives.
Formal epsilon-delta proofs are not assessed in this unit. L’Hôpital’s rule, derivative rules and differentiation tests belong later in the course and are not used to bypass limit reasoning here. Trigonometric limits use radians. Finite graphs and tables provide evidence, not a proof of all nearby behavior. Infinity describes unbounded behavior, not a number to substitute. The Intermediate Value Theorem requires continuity on a closed interval and an intermediate output; it guarantees existence, not uniqueness.
The Organic Chemistry Tutor video title, creator and description were checked; the full video was not reviewed. Khan Academy’s BC unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Section 2.2 was consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. Camera rotation does not add a variable or change slope; use labeled coordinates and axis scales.
Independent teacher review and observation of students remain pending. Checks do not certify scientific accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus BC questions and scoring guides. The archive spans multiple units; no entire exam question is assigned to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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