What are the three checks for continuity?
You will be able to: Justify continuity or identify the failed condition at a point.
What are the three checks for continuity?
A bridge connects two approaching road segments only if there is a point to connect and both sides meet that same location. For a function, the matching quantity is an output height.
A useful starting point: How do holes, jumps and unbounded breaks differ? →
Words and symbols before equations
- Continuous at a
- f(a) exists, the finite limit exists, and they are equal.
- Existence
- The required quantity is defined as a real number.
- Equality check
- Compare the nearby limit with the assigned point value.
- Corner
- A change in slope that need not interrupt continuity.
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. f(x)=x+2 for x≠1 and f(1) is assigned by the slider. Continuity requires the point value to equal the existing limit 3.
Read the picture in three steps
- Read the species and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- The left and right limits equal 3; f(1)=5. The finite limit exists, but the assigned value differs: removable discontinuity. Open circle: branch exclusion; filled point: assigned value.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the chemistry
First determine f(a) from the actual function definition. Do not substitute into an expression that applies only away from a.
Next find the finite limit, usually by checking both sides or applying continuity of a familiar formula. A jump fails this step.
Finally compare the two numbers. Having both a value and a limit is not enough if they disagree.
A sharp corner such as y=abs(x) at zero is continuous because both side limits and the value equal zero. Smoothness and differentiability are separate questions for later units.
A worked example, step by step
For f(x)=x+2 when x≠1 and f(1)=5, apply the continuity checklist at 1.
- f(1)=5 exists.
- The left and right limits of x+2 both equal 3.
- Thus the finite two-sided limit exists and equals 3.
- Because 3≠5, the equality condition fails and f is not continuous at 1.
Continuous does not mean differentiable or visually smooth.
Is abs(x) continuous at zero?
Compare with an explanation
Yes. Both side limits and its assigned value are zero, despite the corner.
Predict. Change one thing. Explain.
Change the point assignment until the three conditions agree. Explain why changing the dot leaves the limiting value fixed.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
The left and right limits equal 3; f(1)=5. The finite limit exists, but the assigned value differs: removable discontinuity. Open circle: branch exclusion; filled point: assigned value.
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. f(x)=x+2 for x≠1 and f(1) is assigned by the slider. Continuity requires the point value to equal the existing limit 3.
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 questionFor h(x)=x² for x≠2 and h(2)=7, apply all three checks and state the repairing value.
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Compare with the answer and four-point rubric
- 1 point: h(2)=7 exists.
- 1 point: The nearby limit is 4.
- 1 point: The equality check fails because 4≠7.
- 1 point: Set h(2)=4 to make it continuous there.
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 1How many checks define continuity at an interior point?
Value exists, finite limit exists, and they agree.
RECALL 2Does a corner necessarily break continuity?
No.
RECALL 3What is the final comparison?
The limit versus f(a).
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
What are the three checks for continuity?
- Continuity at a: f(a) exists; lim as x→a of f(x) exists; limit=f(a).
Remember: Continuous does not mean differentiable or visually smooth.
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. f(x)=x+2 for x≠1 and f(1) is assigned by the slider. Continuity requires the point value to equal the existing limit 3.
Refresh Kid · AP Calculus AB Unit 1 · Objectives LIM-2.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 1.11, 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 current course announcement and linked unit destination were 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 AB 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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