Confirming Continuity over an Interval
Use the domain and one-sided endpoint conditions to state intervals of continuity.
College Board Topic 1.12 · CED p. 44 (PDF p. 49) · CED effective Fall 2020; project course outline 2026–27.
A function is continuous on an interval when it is continuous at every point there. At an included left endpoint use right continuity; at an included right endpoint use left continuity. These one-sided checks reflect the inputs available inside the interval.
Polynomials are continuous for every real x. Rational functions are continuous wherever their denominators are nonzero. Familiar power, exponential, logarithmic and trigonometric functions are continuous on their domains. Always state that domain: ln x needs x>0, √x needs x≥0, and tan x excludes π/2+kπ.
For h(x)=ln(x−1)/(x−4), the domain requires x>1 and x≠4. Continuity holds on (1,4) and (4,∞). A graph may help locate the breaks, but the domain argument explains why there are no other breaks.
Let's work through it
- For h(x)=√(x−1)/(x−3), require x≥1 for the square root and x≠3 for division. Reason: the domain must support each operation.
- The numerator is continuous on its domain and the denominator is continuous. Reason: these are familiar continuous functions.
- State continuous intervals [1,3) and (3,∞), with right continuity at 1. Reason: a quotient is continuous wherever its denominator is nonzero.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Where is (x+1)/(x²−9) continuous?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Where is (x+1)/(x²−9) continuous?
Worked answer and reason
On (−∞,−3), (−3,3), and (3,∞).
Only denominator zeros are excluded.
Retrieval 2
Is ln x continuous on all real numbers?
Worked answer and reason
No. It is continuous on (0,∞).
Its real domain contains only positive inputs.
Catch the tempting mistake
- A familiar function is continuous even outside its domain.
- Continuity is a property at inputs where a function is defined; domain restrictions come first.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 1 problem 19 · No calculator · analytic
- Unit 1 problem 20 · No calculator · analytic
For the teacher
- Discuss one-sided endpoint continuity using √x on [0,4].
- Require domain restrictions before interval notation.
Glossary and deeper reading · 5 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.
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Polynomial
A polynomial in one variable is a finite sum of constant multiples of nonnegative integer powers of that variable. The degree is the highest exponent with a nonzero coefficient. For example, 2x³−x+5 is a polynomial of degree 3. Expressions containing 1/x or √x are not polynomials in x.
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Rational function
A rational function is a quotient of two polynomials. Its domain excludes inputs that make the original denominator zero. For example, (x²−1)/(x−1) agrees with x+1 away from x=1, but remains undefined there. Canceling a common factor does not automatically restore an excluded input.
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Domain
The domain of a function is the set of inputs for which the function is defined. An expression and its context can both restrict the allowed inputs. For example, the real function f(x)=1/(x−2) has domain all real numbers except 2. For a model of elapsed time, a sensible domain may also require t≥0.