AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/1/1-11)
Unit 1 · Topic 1.11
1.11 Defining Continuity at a Point
A function is continuous at x = c when you could draw through that point without lifting your pencil. Formally, three conditions must all hold. On the exam, checking those three conditions in writing is how you earn the point.
Key terms
- continuity
- continuous at a point
- f(c) is defined
- limit exists
- piecewise function
The definition
f is continuous at x = c if all three of these are true:
- f(c) is defined (there is a point on the graph at x = c).
- lim (x→c) f(x) exists (the left-hand and right-hand limits are equal).
- lim (x→c) f(x) = f(c) (the limit and the value match).
Which condition fails
Each type of discontinuity breaks a specific condition. A hole with no dot breaks condition 1. A hole with a dot somewhere else passes conditions 1 and 2 but breaks condition 3. A jump breaks condition 2. An infinite discontinuity breaks condition 2 (and often condition 1).
There is also one-sided continuity. f is continuous from the right at c if lim (x→c⁺) f(x) = f(c), and continuous from the left if lim (x→c⁻) f(x) = f(c). A jump function is often continuous from one side only. This idea is what lets a function be continuous at the endpoints of a closed interval (1.12).
Checking a piecewise function
At a break point c, compute three things: f(c) from whichever piece includes c (look for ≤ or ≥), the left-hand limit from the left piece, and the right-hand limit from the right piece. If all three are the same number, f is continuous at c.
Write the conclusion clearly: “Since lim (x→2⁻) f(x) = lim (x→2⁺) f(x) = f(2) = 5, f is continuous at x = 2.”
Continuity from a graph, a table or other facts
On a graph, continuity at c means the curve comes into the solid dot at x = c from both sides with no gap, hole or jump.
A table alone can't prove continuity, because it only shows a few points. If a problem says the function is continuous, or differentiable (which forces continuity, as you'll see in 2.4), you can use that fact directly.
In a free-response answer, a short form is enough when it shows all three values: “lim (x→c) f(x) = f(c) = 5, so f is continuous at x = c.”
Why continuity matters
Continuity is a condition in many theorems you'll use all year: the Intermediate Value Theorem (1.16), the Mean Value Theorem (5.1) and the Extreme Value Theorem (5.2). Before you can use any of them, you must confirm that the function is continuous. Continuity is also required for differentiability (2.4).
Worked examples
Try each one yourself first, then open the solution.
- Example 1
A continuous piecewise function
f(x) = x² + 1 for x < 2, and f(x) = 4x − 3 for x ≥ 2. Is f continuous at x = 2? Justify.
Show the solutionHide the solution
- Step 1: f(2) uses the second piece (x ≥ 2): f(2) = 8 − 3 = 5. Condition 1 holds.
- Step 2: Left-hand limit: lim (x→2⁻) (x² + 1) = 5. Right-hand limit: lim (x→2⁺) (4x − 3) = 5. They match, so lim (x→2) f(x) = 5. Condition 2 holds.
- Step 3: The limit equals f(2) = 5. Condition 3 holds.
Answer: Yes. Since lim (x→2⁻) f(x) = lim (x→2⁺) f(x) = f(2) = 5, f is continuous at x = 2.
- Example 2
Trap: the limit exists but continuity fails
g(x) = (x² − 1) / (x − 1) for x ≠ 1, and g(1) = 3. Is g continuous at x = 1?
Show the solutionHide the solution
- Step 1: Condition 1: g(1) = 3, so g(1) is defined.
- Step 2: Condition 2: for x ≠ 1, g(x) = (x − 1)(x + 1) / (x − 1) = x + 1, so lim (x→1) g(x) = 2. The limit exists.
- Step 3: Condition 3: 2 ≠ 3, so the limit does not equal the value. This condition fails.
Answer: No. lim (x→1) g(x) = 2 but g(1) = 3, so g is not continuous at x = 1 (a removable discontinuity).
Common mistakes
- Checking only that the left and right limits match. You also need f(c) to exist and to equal that limit.
- Using the wrong piece for f(c). The piece with ≤ or ≥ at c is the one that defines f(c).
- Writing a conclusion with no evidence, like “f is continuous because there's no break.” Show the three values.
On the exam
- Free-response questions often ask “Is f continuous at x = c? Justify your answer.” The point goes to a response that shows the limit and the function value and states that they are equal (or not).
- If a function is given as a graph, you can still use the three conditions: read f(c) from the dot and the limits from the curve.
Connected topics
Videos
Check yourself
4 questions on 1.11 Defining Continuity at a Point. Pick an answer to see if you got it, and why.
Let f be the function with f(x) = x² + 1 for x < 1, f(1) = 3 and f(x) = 3x − 1 for x > 1. Which of the following is true?
Let f(x) = (x³ − 8)/(x − 2) for x ≠ 2, and let f(2) = 8. Which of the following statements are true? I. f(2) exists. II. lim (x→2) f(x) exists. III. f is continuous at x = 2.
The function g is continuous at x = 4, and g(4) = 3. Which of the following must be true?
Let f(x) = kx² − 1 for x ≤ 2, and f(x) = 3x + k for x > 2, where k is a constant. For what value of k is f continuous at x = 2 ?
0 of 4 answered