How do domain restrictions determine continuous intervals?
You will be able to: State continuous intervals and check endpoints correctly.
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 domain restrictions determine continuous intervals?
A rational curve can be continuous throughout each allowed stretch even though some inputs are excluded. Interval notation records which stretches belong to its domain.
A useful starting point: What are the three checks for continuity? →
Words and symbols before equations
- Open interval (a,b)
- Inputs strictly between the endpoints.
- Closed interval [a,b]
- Includes both endpoints.
- Domain restriction
- An input forbidden by the original formula.
- Endpoint continuity
- Use the side that lies within the interval.
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. Real-valued functions. Solid endpoints are included; excluded asymptote lines are dashed. At domain endpoints continuity is understood from within the domain.
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.
- Continuous on (−∞,2) and (2,∞); no value at 2.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Polynomials are continuous on all real inputs. Rational functions are continuous wherever their denominator is nonzero. Logarithms require positive arguments and square roots require nonnegative radicands.
For (x+1)/(x−2), exclude x=2. Its maximal continuous intervals are (−∞,2) and (2,∞).
On a closed interval [a,b], require continuity inside it and the appropriate one-sided agreement at the endpoints: right at a and left at b.
For sqrt(x−1), the domain is [1,∞). It is continuous on this domain, including right-continuity at 1. The absence of negative-radicand inputs is not an interior jump.
A worked example, step by step
State where (x+1)/(x−2) is continuous and whether it is continuous on [0,3].
- The denominator vanishes at x=2.
- All other inputs are valid and the rational function is continuous there.
- The maximal intervals are (−∞,2) and (2,∞).
- It is not continuous on [0,3], which includes the excluded input 2.
Do not restore excluded inputs merely because another expression looks simpler.
Why is 1 included in the domain of sqrt(x−1)?
Compare with an explanation
Its radicand is zero there, and square root of zero is defined.
Predict. Change one thing. Explain.
Switch among a rational function, a square root and a logarithm. Read the allowed intervals and explain the different endpoint brackets.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Continuous on (−∞,2) and (2,∞); no value at 2.
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. Real-valued functions. Solid endpoints are included; excluded asymptote lines are dashed. At domain endpoints continuity is understood from within the domain.
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 questionState the domain and continuity of sqrt(x+3), explaining the left endpoint and what happens below it.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: The radicand requires x≥−3.
- 1 point: The domain is [−3,∞).
- 1 point: The function is continuous there, including right-continuity at −3.
- 1 point: Below −3 the real square root is not defined; those inputs are outside this domain.
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 1Where is a rational function continuous?
At every input with nonzero denominator.
RECALL 2What inequality defines a logarithm’s argument?
Strictly positive.
RECALL 3Why use one side at a closed endpoint?
Only that side lies within the interval being studied.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do domain restrictions determine continuous intervals?
- Check the original domain first.
- Closed-interval continuity uses inward one-sided limits at the endpoints.
Remember: Do not restore excluded inputs merely because another expression looks simpler.
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. Real-valued functions. Solid endpoints are included; excluded asymptote lines are dashed. At domain endpoints continuity is understood from within the domain.
Refresh Kid · AP Calculus BC Unit 1 · Objectives LIM-2.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.12, LIM-2.B. 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.
Want to work through this with a tutor?
Bring your question about How do domain restrictions determine continuous intervals? Your explanation and answers remain free to access.