Calculus
Write your questions
1.4 Continuity Name:_________________________
and thoughts here!
Defining Continuity: Notes
Formal Definition of Continuity:
For to be continuous at , the following three conditions must be met:
1.
2.
3.
Continuous function…or continuous on its domain?
Types of Discontinuities:
1.
2.
3.
For each function identify the x value and type of each discontinuity.
2 2 1, 1
1. 2. √2 3
3. 2, 1 2
2 , 2
Write your questions
1.4 Continuity Notes
and thoughts here!
Finding the Domain
Two scenarios to watch for when looking for a restriction on the domain.
5
1.
1
2. √7 3
Find the domain of each function.
4. 5.
√ √
Intermediate Value Theorem (for continuous functions) ‐ IVT
6. Use the IVT to answer the following questions if 2 5.
a. Find 1 .
b. Find 2 .
c. Find 3 .
d. Does the function have a zero? How do you know?
1.4 Continuity
Calculus Name: _____________________________ Practice
Identify and classify each point of discontinuity of the given function.
1. 𝑓 𝑥 2. 𝑓 𝑥 3. 𝑓 𝑥 4. 𝑓 𝑥 √2 6𝑥
3 2𝑥, 𝑥 2
5. 𝑓 𝑥 6. 𝑓 𝑥 7. 𝑓 𝑥
𝑥 3, 𝑥 2
𝑥
5𝑥 1, 𝑥 1 𝑥2 1
𝑥 4 3, 𝑥 𝑒
8. 𝑓 𝑥 ⎧ 𝑥 11 , 10. 𝑓 𝑥 𝑒
𝑥 3, 𝑥 1 ⎪ ln 𝑥4 , 𝑥 𝑒
9. 𝑓 𝑥 𝑥 3, 𝑥 4
⎨
⎪
⎩ 5, 𝑥 4
Find the domain of each function.
11. 𝑠 𝑥
√ 12. 𝑤 𝑡 13. 𝑓 𝑥
√ √
14. 𝑣 𝑡 15. 𝑔 𝑥 16. 𝑔 𝑤
√ √
17. 𝑠 𝑡 √𝑡 8 √ 19. 𝑔 𝑥 𝑥 11𝑥 30
18. ℎ 𝑡
Below is a table of values for a continuous function 𝑓. Use this table to answer questions 20‐22.
𝑥 3 4 5 6 7
𝑓 𝑥 4 1 3 1 6
20. On the interval 3 𝑥 7, must there be a value of 𝑥 for which 𝑓 𝑥 5? Explain.
21. On the interval 3 𝑥 7, could there be a value of 𝑥 for which 𝑓 𝑥 7? Explain.
22. What is the minimum number of zeros 𝑓 must have on the interval 3 𝑥 7?
Below is a table of values for a continuous function 𝑔. Use this table to answer questions 23‐26.
𝑥 0 2 15 32 50
𝑔 𝑥 1 10 17 10 8
23. On the interval 0 𝑥 15, must there be a value of 𝑥 for which 𝑔 𝑥 3? Explain.
24. On the interval 0 𝑥 50, must there be a value of 𝑥 for which 𝑔 𝑥 11? Explain.
25. What is the minimum number of zeros 𝑔 must have on the interval 15 𝑥 50?
26. What is the minimum number of zeros 𝑔 must have on the interval 0 𝑥 50?
1.4 Continuity Test Prep
1. The graph of the function 𝑓 𝑥 is shown below:
Which of the following statements is true about 𝑓?
I. 𝑓 is undefined at 𝑥 1.
II. 𝑓 is defined but not continuous at 𝑥 2.
III. 𝑓 is defined and continuous at 𝑥 3.
(A) Only I (B) Only II (C) I and II (D) I and III (E) None of the statements
are true.
2. Let 𝑦 . This function has a hole. What is the y-value of the hole?
(A) (B) 3 (C) (D) 0 (E) 3
3. For which value of 𝑘 is the following function continuous at 𝑥 4?
𝜋
sin , 𝑥 4
𝑥
𝑓 𝑥 𝑥
𝑘 , 𝑥 4
2
(A) 𝑘 2 (B) 𝑘 1 (C) 𝑘 1 (D) 𝑘 (E) 𝑘
4.
𝑥 0 1 2
𝑓 𝑥 1 𝑘 2
The function 𝑓 is continuous on the closed interval 0, 2 and has values that are given in the table above.
The equation 𝑔 𝑥 must have at least two intersections with 𝑓 in the interval 0, 2 if 𝑘
(A) 0 (B) (C) 1 (D) 2 (E) 3
𝑥2 𝑘 2𝑥 6
5. For what value of 𝑘 will the function 𝑓 𝑥 have a point discontinuity at 𝑥 𝑘?
𝑥 𝑘
(A) 𝑘 1 (B) 𝑘 0 (C) 𝑘 1 (D) 𝑘 2 (E) 𝑘 3
6. Suppose 𝑓 is continuous on the closed interval 0,4 and suppose 𝑓 0 1, 𝑓 1 2, 𝑓 2 0,
𝑓 3 3, 𝑓 4 3. Which of the following statements about the zeros of 𝑓 on 0,4 is always true?
(A) 𝑓 has exactly one zero on (B) 𝑓 has more than one zero (C) 𝑓 has more than two zeros
0, 4 . on 0, 4 . on 0, 4 .
(D) 𝑓 has exactly two zeros on (E) None of the statements
0, 4 . above is true.
Questions 7 through 9 are based on the function f (x) shown in the graph below:
y
7. The function 𝑓 𝑥 has a removable discontinuity at:
(A) 𝑥 2 only (B) 𝑥 0 only (C) 𝑥 1 only
(D) 𝑥 2 and 𝑥 0 only (E) 𝑓 𝑥 has no removable discontinuities.
8. On what intervals is 𝑓 𝑥 continuous?
(A) 3, 2 ∪ 2, 0 ∪ 0, 2.5 (B) 3, 2 ∪ 2, 0 ∪ 0, 2.5
(C) 3, 2 ∪ 2, 0 ∪ 0, 2.5 (D) 3, 2 ∪ 2, 0 ∪ 0, 2.5
(E) 3, 2 ∪ 2, 0 ∪ 0, 1 ∪ 1, 2.5
9. The function has a jump discontinuity at:
(A) 𝑥 2 only (B) 𝑥 0 only (C) 𝑥 1 only
(D) 𝑥 2 and 𝑥 0 only (E) 𝑓 𝑥 has no jump discontinuities.
FREE RESPONSE ON THE BACK!
For this Free Response problem, answer each question as completely as possible. Do NOT look at the
answers until completed! When done, use the Solution Key to grade your work. Put your score in the
box below.
The graph of a function 𝑓 is shown below and describes the position of a particle as it moves along the 𝑦-
axis with respect to time.
y
xt
a. Describe the movement of the particle on the interval 1,3 .
b. Assume 𝑓 𝑡 1 for 𝑡 6, and 𝑦 1 is an asymptote. Describe the movement of the particle as
𝑡 approaches infinity.
c. Can we use the Intermediate Value Theorem on the interval 1,2 to show that 𝑓 has a zero in
that interval? On the interval 2,5 ? Explain your reasoning.
