Unit 1: Limits and ContinuityContinuity on Intervals

Concept 4.220Built for May 2027 AP Exam

Continuity on Intervals

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

Continuity at a point extends to continuity on an interval, and the extension is where most of the bookkeeping lives. On an open interval, continuity means continuity at every interior point. On a closed interval, it means continuity at every interior point plus continuity from the right at the left endpoint and from the left at the right endpoint. Half-open intervals take only the endpoint rule their bracket permits. None of this changes the function; it changes what you are required to check. The practical version of the skill is to read a formula, locate the places where it could fail, and then write the answer as a union of intervals with the correct bracket at every boundary.

On the AP Exam

AP Calculus BC tests this skill through the representations and reasoning described below.

What you’ll see: "State the interval on which $f$ is continuous", a radical whose argument vanishes at an endpoint, a logarithm whose argument vanishes, or a rational function with a hole punched out of an otherwise continuous domain.
What you’ll do: Find the domain, test the interior, apply the correct one-sided rule at each endpoint, and write the result with interval notation that matches the behavior you verified.
Memory hook: Inside needs both sides, the ends need one.
Use it on the exam: For MCQs: Build the interval from the domain first and then adjust the brackets. Most wrong choices differ only in a bracket, which makes this a notation question as much as a calculus one. For FRQs: The rubric usually separates the interior statement from the endpoint statement, so write them as two sentences rather than folding them into one. On released BC papers: No released BC free-response part from 2017 to 2026 asks for an interval of continuity as its own demand. The interval language does appear inside stems, where a function is introduced as continuous on a stated closed interval before its graph is used, but that is a given rather than a task scored under this topic. The multiple-choice sections, by contrast, use the interval formulation regularly, because a bracket is an efficient distractor: four choices that differ only at the endpoints test the definition precisely and grade in one second. That is the realistic shape of the item you will meet.

Concept Review

Interior, Then the Ends

Figure 1 — The interior is checked point by point; the endpoints only from the inside.

Caption. The curve is continuous on the open interval and the two filled endpoints mark the closed extension. At each end, only the side that belongs to the domain is available to test, which is why the endpoint rules are one-sided.

The three definitions, stated exactly:

Continuous on $(a, b)$ : continuous at every $c$ with $a < c < b$ .
Continuous on $[a, b]$ : continuous on $(a, b)$ , continuous from the right at $a$ , and continuous from the left at $b$ .
Continuous on $[a, b)$ or $(a, b]$ : the interior, plus the single endpoint rule the closed end permits.

A continuous curve drawn from a to b with filled endpoints, one-sided continuity labels at each end, and a reference card listing what open, closed, and half-open intervals require

What Each Interval Type Requires

Interval	Interior	At the Left End	At the Right End
$(a, b)$	Required	Not required	Not required
$[a, b]$	Required	Right-continuity required	Left-continuity required
$[a, b)$	Required	Right-continuity required	Not required
$(a, b]$	Required	Not required	Left-continuity required
$(a, \infty)$	Required	Not required	No endpoint to test
$[a, \infty)$	Required	Right-continuity required	No endpoint to test

The bracket is not decoration. It is the statement of which one-sided limit the definition demands.

Reading the Interval Off the Formula

Every elementary function is continuous on its domain, so the interval question is a domain question plus an endpoint check:

Polynomials: continuous on $(-\infty, \infty)$ .
Rational functions: continuous on their domain, which is the real line with the zeros of the denominator removed. Those removals are open brackets, because the function is undefined there.
Even radicals: continuous where the radicand is positive, and right- or left-continuous at a zero radicand depending on which side the radicand is non-negative. The zero itself belongs to the interval with a closed bracket.
Logarithms: continuous where the argument is positive. The zero of the argument is excluded with an open bracket, because the logarithm is undefined there.

Compositions and Unions

Continuity on intervals survives the operations of the previous topic: sums, products, quotients, and compositions of functions continuous on intervals are continuous wherever they are defined. That is why a single expression such as $\frac{\sqrt{x-2}}{x-5}$ can be certified by examining two features, the radicand and the denominator, rather than by testing points one at a time.

The result is typically a union of intervals, and each piece carries its own brackets. Writing the union correctly is part of the answer, not a formality.

Common Misconception Check

Myth 1: "Continuous on $[a, b]$ means continuous at $a$ in the two-sided sense."
- Truth: Only right-continuity is required at $a$ , because the function is not defined to the left of $a$ . Demanding a two-sided limit is how a correct function gets declared discontinuous.
Myth 2: "An endpoint can be included whenever the curve touches it."
- Truth: Inclusion depends on the bracket, which depends on the function's domain, not on how the graph looks.
Myth 3: "If a function is continuous on $(a, b)$ , it is continuous on $[a, b]$ ."
- Truth: The closed interval adds two one-sided requirements that the open interval never tested.

Topic-Specific Study Method

Read the Bracket Before the Formula

Borrowed idea: Where does continuity actually get asserted? On an interval, which means at every point inside it and, when the interval is closed, at the endpoints from the inside. Most students check the interior and forget the endpoints. The endpoints are where radicals and logarithms live, so they are exactly the points that need the one-sided test. Does the interval's bracket change the answer? It changes the requirement, not the function. A closed bracket asks for the one-sided limit; an open parenthesis asks for nothing. Read the interval notation before the formula, because the bracket tells you which side of each endpoint is even available to test.

Exam Scenario	What the Page Shows	What the Prompt Is Really Asking	Required Execution
"State the interval of continuity"	A formula with a radical and a denominator	What is the domain, and which ends are closed?	Domain first, then brackets, then write the union
Even radical	$\sqrt{x - c}$	Is the endpoint included, and from which side?	Right-continuous at $c$ , closed bracket at $c$
Logarithm	$\ln(x - c)$	Is the endpoint included?	Argument positive, open bracket at $c$
Rational with a hole	Denominator zero at $c$	Is $c$ excluded and with which bracket?	Open bracket on both sides of $c$
Composition	$\sqrt{x}$ inside another function	Does the inner range stay in the outer domain?	Test the inner function's range, then the outer's domain

Executable Action Steps: ① Find the domain before saying anything about continuity. The interval of continuity is always a subset of the domain, so an interval that includes a point outside the domain is wrong on its face. ② At each finite endpoint, compute the one-sided limit and compare it to the value. That single comparison is the endpoint rule, and writing it out prevents bracket errors. ③ Write the final answer as a union of intervals with brackets justified one at a time. If you cannot say why a bracket is closed, it probably is not.

Training & Error Diagnosis

Core Archetype (Simplest Form): State the interval on which $f(x) = \sqrt{x - 3}$ is continuous.

Standard Execution:

The domain requires $x - 3 \ge 0$ , so the domain is $[3, \infty)$ .

On $(3, \infty)$ the radicand is positive and the square root is continuous at every such point.

At the endpoint $3$ , only the right-hand limit exists, and $\lim_{x \to 3^+}\sqrt{x-3} = 0 = f(3)$ .

Therefore $f$ is continuous on $[3, \infty)$ , with a closed bracket at $3$ because the endpoint rule holds.

Exam Variation (Context / Parameter Shift): 1. Two Features at Once. State the interval of continuity of $g(x) = \dfrac{\sqrt{x - 2}}{x - 5}$ .

Solution: The radicand requires $x \ge 2$ and the denominator requires $x \neq 5$ . On $(2, 5) \cup (5, \infty)$ the function is a quotient of continuous functions with a nonzero denominator. At $x = 2$ the right-hand limit is $0 = g(2)$ , so the bracket closes. The answer is $[2, 5) \cup (5, \infty)$ .

2. A Logarithm. State the interval of continuity of $h(x) = \ln(4 - x)$ .

Solution: The argument must be positive, so $x < 4$ and the domain is $(-\infty, 4)$ . At $x = 4$ the logarithm is undefined and the limit is $-\infty$ , so the bracket stays open. The answer is $(-\infty, 4)$ .

Common Traps & Failure Signals:

The closed bracket at an excluded point (Signal: your interval includes a zero of the denominator): Excluded points take parentheses on both sides, without exception.
The two-sided demand at an endpoint (Signal: you concluded "not continuous" at the left end of a closed interval): At an endpoint only the interior side is required.
The interior-only answer (Signal: your interval is open when the function is defined and one-sided-continuous at the end): An open bracket where continuity from the inside holds is as wrong as a closed one at a hole.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Interior Only: You checked the middle and skipped both ends.
- Bracket Ignored: Because you treated a closed endpoint like an interior point, you demanded a two-sided limit that the domain never provides, and then reported a continuity failure that was really a notation error.
- Half-Open Blurred: You collapsed the four interval types into one rule.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; No part asks for an interval of continuity; stems state the interval before using the graph
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; Zero parts, with the interval language appearing as a given rather than a task
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

Two Features

On which interval is $f(x) = \dfrac{\sqrt{x - 2}}{x - 5}$ continuous?

MCQ 2

The Endpoint Rule

The function $g(x) = \sqrt{7 - x}$ is continuous on which interval?

MCQ 3

Which Requirement Applies?

If $f$ is continuous on $[2, 6]$ , which statement must be true?

FRQ 4

Intervals, Endpoints, and Justification

Let $f(x) = \dfrac{\sqrt{x + 1}}{x - 3}$ .

(a) Determine the domain of $f$ .

(b) State the interval or intervals on which $f$ is continuous, justifying the bracket at every finite endpoint.

(c) Explain why $f$ is not continuous at $x = 3$ , and state which clause of the definition fails there.

Cheatsheet

Core Reference

	Interior Points	Left Endpoint	Right Endpoint
$(a, b)$	Required	None	None
$[a, b]$	Required	Right-continuity	Left-continuity
$[a, b)$	Required	Right-continuity	None
$(a, b]$	Required	None	Left-continuity
$[a, \infty)$	Required	Right-continuity	No endpoint
$(-\infty, b]$	Required	No endpoint	Left-continuity

Sentences That Score

TOPIC REVIEW • NEXT STEP IN YOUR LEARNING LOOP

Ready to check your understanding?

Strengthen this topic with AP-style practice, or continue to next topic.

Start Topic Practice Next Topic: 1.17 Polynomial and Rational Functions

Unit 1: Limits and ContinuityContinuity on Intervals

Concept 4.220Built for May 2027 AP Exam

Continuity on Intervals

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

AP Calculus BC tests this skill through the representations and reasoning described below.

What you’ll see: "State the interval on which $f$ is continuous", a radical whose argument vanishes at an endpoint, a logarithm whose argument vanishes, or a rational function with a hole punched out of an otherwise continuous domain.
What you’ll do: Find the domain, test the interior, apply the correct one-sided rule at each endpoint, and write the result with interval notation that matches the behavior you verified.
Memory hook: Inside needs both sides, the ends need one.
Use it on the exam: For MCQs: Build the interval from the domain first and then adjust the brackets. Most wrong choices differ only in a bracket, which makes this a notation question as much as a calculus one. For FRQs: The rubric usually separates the interior statement from the endpoint statement, so write them as two sentences rather than folding them into one. On released BC papers: No released BC free-response part from 2017 to 2026 asks for an interval of continuity as its own demand. The interval language does appear inside stems, where a function is introduced as continuous on a stated closed interval before its graph is used, but that is a given rather than a task scored under this topic. The multiple-choice sections, by contrast, use the interval formulation regularly, because a bracket is an efficient distractor: four choices that differ only at the endpoints test the definition precisely and grade in one second. That is the realistic shape of the item you will meet.

Concept Review

Interior, Then the Ends

Figure 1 — The interior is checked point by point; the endpoints only from the inside.

The three definitions, stated exactly:

Continuous on $(a, b)$ : continuous at every $c$ with $a < c < b$ .
Continuous on $[a, b]$ : continuous on $(a, b)$ , continuous from the right at $a$ , and continuous from the left at $b$ .
Continuous on $[a, b)$ or $(a, b]$ : the interior, plus the single endpoint rule the closed end permits.

A continuous curve drawn from a to b with filled endpoints, one-sided continuity labels at each end, and a reference card listing what open, closed, and half-open intervals require

What Each Interval Type Requires

Interval	Interior	At the Left End	At the Right End
$(a, b)$	Required	Not required	Not required
$[a, b]$	Required	Right-continuity required	Left-continuity required
$[a, b)$	Required	Right-continuity required	Not required
$(a, b]$	Required	Not required	Left-continuity required
$(a, \infty)$	Required	Not required	No endpoint to test
$[a, \infty)$	Required	Right-continuity required	No endpoint to test

The bracket is not decoration. It is the statement of which one-sided limit the definition demands.

Reading the Interval Off the Formula

Every elementary function is continuous on its domain, so the interval question is a domain question plus an endpoint check:

Polynomials: continuous on $(-\infty, \infty)$ .
Rational functions: continuous on their domain, which is the real line with the zeros of the denominator removed. Those removals are open brackets, because the function is undefined there.
Even radicals: continuous where the radicand is positive, and right- or left-continuous at a zero radicand depending on which side the radicand is non-negative. The zero itself belongs to the interval with a closed bracket.
Logarithms: continuous where the argument is positive. The zero of the argument is excluded with an open bracket, because the logarithm is undefined there.

Compositions and Unions

The result is typically a union of intervals, and each piece carries its own brackets. Writing the union correctly is part of the answer, not a formality.

Common Misconception Check

Myth 1: "Continuous on $[a, b]$ means continuous at $a$ in the two-sided sense."
- Truth: Only right-continuity is required at $a$ , because the function is not defined to the left of $a$ . Demanding a two-sided limit is how a correct function gets declared discontinuous.
Myth 2: "An endpoint can be included whenever the curve touches it."
- Truth: Inclusion depends on the bracket, which depends on the function's domain, not on how the graph looks.
Myth 3: "If a function is continuous on $(a, b)$ , it is continuous on $[a, b]$ ."
- Truth: The closed interval adds two one-sided requirements that the open interval never tested.

Topic-Specific Study Method

Read the Bracket Before the Formula

Borrowed idea: Where does continuity actually get asserted? On an interval, which means at every point inside it and, when the interval is closed, at the endpoints from the inside. Most students check the interior and forget the endpoints. The endpoints are where radicals and logarithms live, so they are exactly the points that need the one-sided test. Does the interval's bracket change the answer? It changes the requirement, not the function. A closed bracket asks for the one-sided limit; an open parenthesis asks for nothing. Read the interval notation before the formula, because the bracket tells you which side of each endpoint is even available to test.

Exam Scenario	What the Page Shows	What the Prompt Is Really Asking	Required Execution
"State the interval of continuity"	A formula with a radical and a denominator	What is the domain, and which ends are closed?	Domain first, then brackets, then write the union
Even radical	$\sqrt{x - c}$	Is the endpoint included, and from which side?	Right-continuous at $c$ , closed bracket at $c$
Logarithm	$\ln(x - c)$	Is the endpoint included?	Argument positive, open bracket at $c$
Rational with a hole	Denominator zero at $c$	Is $c$ excluded and with which bracket?	Open bracket on both sides of $c$
Composition	$\sqrt{x}$ inside another function	Does the inner range stay in the outer domain?	Test the inner function's range, then the outer's domain

Executable Action Steps: ① Find the domain before saying anything about continuity. The interval of continuity is always a subset of the domain, so an interval that includes a point outside the domain is wrong on its face. ② At each finite endpoint, compute the one-sided limit and compare it to the value. That single comparison is the endpoint rule, and writing it out prevents bracket errors. ③ Write the final answer as a union of intervals with brackets justified one at a time. If you cannot say why a bracket is closed, it probably is not.

Training & Error Diagnosis

Core Archetype (Simplest Form): State the interval on which $f(x) = \sqrt{x - 3}$ is continuous.

Standard Execution:

The domain requires $x - 3 \ge 0$ , so the domain is $[3, \infty)$ .

On $(3, \infty)$ the radicand is positive and the square root is continuous at every such point.

At the endpoint $3$ , only the right-hand limit exists, and $\lim_{x \to 3^+}\sqrt{x-3} = 0 = f(3)$ .

Therefore $f$ is continuous on $[3, \infty)$ , with a closed bracket at $3$ because the endpoint rule holds.

Exam Variation (Context / Parameter Shift): 1. Two Features at Once. State the interval of continuity of $g(x) = \dfrac{\sqrt{x - 2}}{x - 5}$ .

Solution: The radicand requires $x \ge 2$ and the denominator requires $x \neq 5$ . On $(2, 5) \cup (5, \infty)$ the function is a quotient of continuous functions with a nonzero denominator. At $x = 2$ the right-hand limit is $0 = g(2)$ , so the bracket closes. The answer is $[2, 5) \cup (5, \infty)$ .

2. A Logarithm. State the interval of continuity of $h(x) = \ln(4 - x)$ .

Solution: The argument must be positive, so $x < 4$ and the domain is $(-\infty, 4)$ . At $x = 4$ the logarithm is undefined and the limit is $-\infty$ , so the bracket stays open. The answer is $(-\infty, 4)$ .

Common Traps & Failure Signals:

The closed bracket at an excluded point (Signal: your interval includes a zero of the denominator): Excluded points take parentheses on both sides, without exception.
The two-sided demand at an endpoint (Signal: you concluded "not continuous" at the left end of a closed interval): At an endpoint only the interior side is required.
The interior-only answer (Signal: your interval is open when the function is defined and one-sided-continuous at the end): An open bracket where continuity from the inside holds is as wrong as a closed one at a hole.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Interior Only: You checked the middle and skipped both ends.
- Bracket Ignored: Because you treated a closed endpoint like an interior point, you demanded a two-sided limit that the domain never provides, and then reported a continuity failure that was really a notation error.
- Half-Open Blurred: You collapsed the four interval types into one rule.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; No part asks for an interval of continuity; stems state the interval before using the graph
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; Zero parts, with the interval language appearing as a given rather than a task
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

Two Features

On which interval is $f(x) = \dfrac{\sqrt{x - 2}}{x - 5}$ continuous?

MCQ 2

The Endpoint Rule

The function $g(x) = \sqrt{7 - x}$ is continuous on which interval?

MCQ 3

Which Requirement Applies?

If $f$ is continuous on $[2, 6]$ , which statement must be true?

FRQ 4

Intervals, Endpoints, and Justification

Let $f(x) = \dfrac{\sqrt{x + 1}}{x - 3}$ .

(a) Determine the domain of $f$ .

(b) State the interval or intervals on which $f$ is continuous, justifying the bracket at every finite endpoint.

(c) Explain why $f$ is not continuous at $x = 3$ , and state which clause of the definition fails there.

Cheatsheet

Core Reference

	Interior Points	Left Endpoint	Right Endpoint
$(a, b)$	Required	None	None
$[a, b]$	Required	Right-continuity	Left-continuity
$[a, b)$	Required	Right-continuity	None
$(a, b]$	Required	None	Left-continuity
$[a, \infty)$	Required	Right-continuity	No endpoint
$(-\infty, b]$	Required	No endpoint	Left-continuity

Sentences That Score

TOPIC REVIEW • NEXT STEP IN YOUR LEARNING LOOP

Ready to check your understanding?

Strengthen this topic with AP-style practice, or continue to next topic.

Start Topic Practice Next Topic: 1.17 Polynomial and Rational Functions

Unit 1: Limits and ContinuityContinuity on Intervals

Concept 4.220Built for May 2027 AP Exam

Continuity on Intervals

Reviewed by ACEpath AP Math Editorial Team

Last Updated: September 2026

30-second summary

On the AP Exam

AP Calculus BC tests this skill through the representations and reasoning described below.

What you’ll see: "State the interval on which $f$ is continuous", a radical whose argument vanishes at an endpoint, a logarithm whose argument vanishes, or a rational function with a hole punched out of an otherwise continuous domain.
What you’ll do: Find the domain, test the interior, apply the correct one-sided rule at each endpoint, and write the result with interval notation that matches the behavior you verified.
Memory hook: Inside needs both sides, the ends need one.
Use it on the exam: For MCQs: Build the interval from the domain first and then adjust the brackets. Most wrong choices differ only in a bracket, which makes this a notation question as much as a calculus one. For FRQs: The rubric usually separates the interior statement from the endpoint statement, so write them as two sentences rather than folding them into one. On released BC papers: No released BC free-response part from 2017 to 2026 asks for an interval of continuity as its own demand. The interval language does appear inside stems, where a function is introduced as continuous on a stated closed interval before its graph is used, but that is a given rather than a task scored under this topic. The multiple-choice sections, by contrast, use the interval formulation regularly, because a bracket is an efficient distractor: four choices that differ only at the endpoints test the definition precisely and grade in one second. That is the realistic shape of the item you will meet.

Concept Review

Interior, Then the Ends

Figure 1 — The interior is checked point by point; the endpoints only from the inside.

The three definitions, stated exactly:

Continuous on $(a, b)$ : continuous at every $c$ with $a < c < b$ .
Continuous on $[a, b]$ : continuous on $(a, b)$ , continuous from the right at $a$ , and continuous from the left at $b$ .
Continuous on $[a, b)$ or $(a, b]$ : the interior, plus the single endpoint rule the closed end permits.

A continuous curve drawn from a to b with filled endpoints, one-sided continuity labels at each end, and a reference card listing what open, closed, and half-open intervals require

What Each Interval Type Requires

Interval	Interior	At the Left End	At the Right End
$(a, b)$	Required	Not required	Not required
$[a, b]$	Required	Right-continuity required	Left-continuity required
$[a, b)$	Required	Right-continuity required	Not required
$(a, b]$	Required	Not required	Left-continuity required
$(a, \infty)$	Required	Not required	No endpoint to test
$[a, \infty)$	Required	Right-continuity required	No endpoint to test

The bracket is not decoration. It is the statement of which one-sided limit the definition demands.

Reading the Interval Off the Formula

Every elementary function is continuous on its domain, so the interval question is a domain question plus an endpoint check:

Polynomials: continuous on $(-\infty, \infty)$ .
Rational functions: continuous on their domain, which is the real line with the zeros of the denominator removed. Those removals are open brackets, because the function is undefined there.
Even radicals: continuous where the radicand is positive, and right- or left-continuous at a zero radicand depending on which side the radicand is non-negative. The zero itself belongs to the interval with a closed bracket.
Logarithms: continuous where the argument is positive. The zero of the argument is excluded with an open bracket, because the logarithm is undefined there.

Compositions and Unions

The result is typically a union of intervals, and each piece carries its own brackets. Writing the union correctly is part of the answer, not a formality.

Common Misconception Check

Myth 1: "Continuous on $[a, b]$ means continuous at $a$ in the two-sided sense."
- Truth: Only right-continuity is required at $a$ , because the function is not defined to the left of $a$ . Demanding a two-sided limit is how a correct function gets declared discontinuous.
Myth 2: "An endpoint can be included whenever the curve touches it."
- Truth: Inclusion depends on the bracket, which depends on the function's domain, not on how the graph looks.
Myth 3: "If a function is continuous on $(a, b)$ , it is continuous on $[a, b]$ ."
- Truth: The closed interval adds two one-sided requirements that the open interval never tested.

Topic-Specific Study Method

Read the Bracket Before the Formula

Borrowed idea: Where does continuity actually get asserted? On an interval, which means at every point inside it and, when the interval is closed, at the endpoints from the inside. Most students check the interior and forget the endpoints. The endpoints are where radicals and logarithms live, so they are exactly the points that need the one-sided test. Does the interval's bracket change the answer? It changes the requirement, not the function. A closed bracket asks for the one-sided limit; an open parenthesis asks for nothing. Read the interval notation before the formula, because the bracket tells you which side of each endpoint is even available to test.

Exam Scenario	What the Page Shows	What the Prompt Is Really Asking	Required Execution
"State the interval of continuity"	A formula with a radical and a denominator	What is the domain, and which ends are closed?	Domain first, then brackets, then write the union
Even radical	$\sqrt{x - c}$	Is the endpoint included, and from which side?	Right-continuous at $c$ , closed bracket at $c$
Logarithm	$\ln(x - c)$	Is the endpoint included?	Argument positive, open bracket at $c$
Rational with a hole	Denominator zero at $c$	Is $c$ excluded and with which bracket?	Open bracket on both sides of $c$
Composition	$\sqrt{x}$ inside another function	Does the inner range stay in the outer domain?	Test the inner function's range, then the outer's domain

Executable Action Steps: ① Find the domain before saying anything about continuity. The interval of continuity is always a subset of the domain, so an interval that includes a point outside the domain is wrong on its face. ② At each finite endpoint, compute the one-sided limit and compare it to the value. That single comparison is the endpoint rule, and writing it out prevents bracket errors. ③ Write the final answer as a union of intervals with brackets justified one at a time. If you cannot say why a bracket is closed, it probably is not.

Training & Error Diagnosis

Core Archetype (Simplest Form): State the interval on which $f(x) = \sqrt{x - 3}$ is continuous.

Standard Execution:

The domain requires $x - 3 \ge 0$ , so the domain is $[3, \infty)$ .

On $(3, \infty)$ the radicand is positive and the square root is continuous at every such point.

At the endpoint $3$ , only the right-hand limit exists, and $\lim_{x \to 3^+}\sqrt{x-3} = 0 = f(3)$ .

Therefore $f$ is continuous on $[3, \infty)$ , with a closed bracket at $3$ because the endpoint rule holds.

Exam Variation (Context / Parameter Shift): 1. Two Features at Once. State the interval of continuity of $g(x) = \dfrac{\sqrt{x - 2}}{x - 5}$ .

Solution: The radicand requires $x \ge 2$ and the denominator requires $x \neq 5$ . On $(2, 5) \cup (5, \infty)$ the function is a quotient of continuous functions with a nonzero denominator. At $x = 2$ the right-hand limit is $0 = g(2)$ , so the bracket closes. The answer is $[2, 5) \cup (5, \infty)$ .

2. A Logarithm. State the interval of continuity of $h(x) = \ln(4 - x)$ .

Solution: The argument must be positive, so $x < 4$ and the domain is $(-\infty, 4)$ . At $x = 4$ the logarithm is undefined and the limit is $-\infty$ , so the bracket stays open. The answer is $(-\infty, 4)$ .

Common Traps & Failure Signals:

The closed bracket at an excluded point (Signal: your interval includes a zero of the denominator): Excluded points take parentheses on both sides, without exception.
The two-sided demand at an endpoint (Signal: you concluded "not continuous" at the left end of a closed interval): At an endpoint only the interior side is required.
The interior-only answer (Signal: your interval is open when the function is defined and one-sided-continuous at the end): An open bracket where continuity from the inside holds is as wrong as a closed one at a hole.

6D Cognitive Error Diagnosis (Where Your Thinking Broke):

Interior Only: You checked the middle and skipped both ends.
- Bracket Ignored: Because you treated a closed endpoint like an interior point, you demanded a two-sided limit that the domain never provides, and then reported a continuity failure that was really a notation error.
- Half-Open Blurred: You collapsed the four interval types into one rule.

Past 10-Year AP Exam Appearance

Testing Frequency: MCQ Only; No part asks for an interval of continuity; stems state the interval before using the graph
10-Year Total Count: 0 of 9 Exams · 0 FRQ Parts; Zero parts, with the interval language appearing as a given rather than a task
FRQ Years Appeared: None in Released FRQs

Practice Examples

MCQ 1

Two Features

On which interval is $f(x) = \dfrac{\sqrt{x - 2}}{x - 5}$ continuous?

MCQ 2

The Endpoint Rule

The function $g(x) = \sqrt{7 - x}$ is continuous on which interval?

MCQ 3

Which Requirement Applies?

If $f$ is continuous on $[2, 6]$ , which statement must be true?

FRQ 4

Intervals, Endpoints, and Justification

Let $f(x) = \dfrac{\sqrt{x + 1}}{x - 3}$ .

(a) Determine the domain of $f$ .

(b) State the interval or intervals on which $f$ is continuous, justifying the bracket at every finite endpoint.

(c) Explain why $f$ is not continuous at $x = 3$ , and state which clause of the definition fails there.

Cheatsheet

Core Reference

	Interior Points	Left Endpoint	Right Endpoint
$(a, b)$	Required	None	None
$[a, b]$	Required	Right-continuity	Left-continuity
$[a, b)$	Required	Right-continuity	None
$(a, b]$	Required	None	Left-continuity
$[a, \infty)$	Required	Right-continuity	No endpoint
$(-\infty, b]$	Required	No endpoint	Left-continuity

Sentences That Score

TOPIC REVIEW • NEXT STEP IN YOUR LEARNING LOOP

Ready to check your understanding?

Strengthen this topic with AP-style practice, or continue to next topic.

Start Topic Practice Next Topic: 1.17 Polynomial and Rational Functions