Defining Continuity at a Point
Check that the function value, finite limit, and equality of the two all hold at a point.
College Board Topic 1.11 · CED p. 43 (PDF p. 48) · CED effective Fall 2020; project course outline 2026–27.
Continuity at x=c has three checks: f(c) exists; lim(x→c)f(x) exists as a finite number; and that limit equals f(c). Each check can fail separately. List the checks explicitly instead of relying on “the graph looks connected.”
A polynomial passes all three checks at every real input because substitution gives its value and its limit. At a piecewise boundary, find f(c) from the branch that includes c and find the one-sided limits from the nearby branches. The branch containing c does not dictate both limiting values.
For f(x)=x+2 when x≠1 and f(1)=9, the value exists and the limit exists, but 3≠9. The function is discontinuous at 1. If f(1) were 3, all three checks would pass.
Let's work through it
- Read f(1)=9 from the special target rule. Reason: continuity includes the actual value.
- Use x+2 nearby to get lim(x→1)f(x)=3. Reason: the nearby expression is continuous.
- Since 3≠9, f is not continuous at 1. Reason: equality of value and limit is the failed condition.
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
If f(c) is undefined but the limit is 4, is f continuous at c?
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
If f(c) is undefined but the limit is 4, is f continuous at c?
Worked answer and reason
No.
The first continuity condition fails.
Retrieval 2
If f(c)=4 and both one-sided limits are 4, is f continuous at c?
Worked answer and reason
Yes.
The value, finite limit and equality conditions all hold.
Catch the tempting mistake
- Existence of a limit is enough for continuity.
- Continuity also requires an actual value equal to that limit.
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
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · 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 17 · No calculator · analytic
- Unit 1 problem 18 · No calculator · analytic
For the teacher
- Use a three-column checklist: value, limit, equality.
- Give examples in which each separate condition is the one that fails.
Glossary and deeper reading · 4 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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Limit
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound. For an input approaching a finite target, its value at that exact input does not determine the limit. For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1. A finite two-sided limit requires the left and right approaches to agree.
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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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One-sided limit
A one-sided limit uses inputs approaching a target from just one side. A left-hand limit uses smaller inputs, and a right-hand limit uses larger inputs. For example, a function that equals 0 left of 2 and 1 right of 2 has different one-sided limits at 2. The two-sided limit exists as a finite real number only when both one-sided limits exist and agree.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.